[{"data":1,"prerenderedAt":1096},["ShallowReactive",2],{"layer:patterns:deepen":3},{"layer":4,"contentHash":1074,"dependencyHashes":1075,"approval":1089,"releaseId":1095},{"schemaVersion":5,"conceptId":6,"locale":7,"depth":8,"revision":5,"title":9,"subtitle":10,"summary":11,"objectives":12,"estimatedMinutes":18,"plate":19,"blocks":40,"sourceIds":1069,"reviewStatus":1070,"authoring":1071},1,"patterns","en","deepen","Why patterns work: rules, algebra and proof","nth terms, equivalent expressions, picture proofs, Pingala’s rhythms, Meru Prastara and patterns that break","Turn rules into algebra and prove them: why the step becomes the coefficient of n, why odd numbers make squares, sums of powers and cubes, the Indian discovery of the Fibonacci numbers and Meru Prastara, why digit patterns stop, and why patterns that look certain can break.",[13,14,15,16,17],"Write and check nth-term formulas for arithmetic and geometric sequences, and decide whether a number is a term.","Explain why the common difference becomes the coefficient of n, and show that different counting pictures give equivalent expressions.","Use differences to tell linear, quadratic and exponential sequences apart, and prove classic sums with pictures and pairing.","Explain the Indian origins of the Fibonacci numbers and Meru Prastara and find patterns inside the triangle.","Give examples of patterns that break and explain why proof matters.",60,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Go deeper",{"label":26,"value":27},"Reading time","≈ 60 minutes",{"label":29,"value":30},"Prior knowledge","Investigate: odd sums, Gauss, conjectures",{"label":32,"value":33},"Chapters","10",{"label":35,"value":36},"Labs","Pattern machine ×3, match pairs, sort game",{"label":38,"value":39},"Key idea","Tₙ = dn + (a − d)",[41,45,51,57,60,82,94,148,153,158,161,170,220,225,228,236,266,271,276,279,290,295,300,303,326,344,348,359,373,462,467,470,519,523,560,563,575,613,618,621,654,702,714,719,722,731,736,739,755,799,803,808,813,886,1021,1025,1031,1036,1040,1056],{"id":42,"type":43,"markdown":44},"intro","prose","So far you have found rules, tested them and used them. This layer asks the deeper question: **why** do the rules work, and how can we be **sure**?\n\nYou will turn rules in words into **algebra**, the language that lets one short line describe infinitely many cases. You will prove that the step of an arithmetic sequence always becomes the number in front of n, that the odd numbers really do add up to squares for ever, and that 1 + 2 + … + n is always n(n + 1) ÷ 2. You will meet the Indian poets and scholars who counted rhythms and discovered the “Fibonacci” numbers and the “Pascal” triangle centuries before the Europeans whose names they carry. And you will meet patterns that look perfect and then break, which is exactly why proof matters.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"how-to","callout","observation","How to use this layer","This is the hardest layer so far. Read each proof slowly with a pencil, and try the small cases yourself. If a line of algebra feels like magic, check it by putting in n = 1, 2 and 3. Nothing in mathematics should be taken on trust, including this page.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch01","chapter","From words to algebra: the nth term","Chapter 01","1 Words to algebra",{"id":58,"type":43,"markdown":59},"algebra","In earlier layers we wrote rules like “multiply the position by 3 and add 1”. Mathematicians shorten this with a **variable**, a letter that stands for any number. With **n** for the position, the rule becomes **3 × n + 1**, often written even more briefly as **3n + 1** (a number written right in front of a letter means multiply).\n\nA rule like 3n + 1 is an **expression**. When we say that the term **T** in position n is given by it, we write a **formula**: **Tₙ = 3n + 1** (say “T sub n equals three n plus one”). The number multiplying n (here 3) is the **coefficient**; the number added on its own (here 1) is the **constant**.\n\nThis small step, using a letter for “any number”, is the beginning of **algebra**. It lets one line say something about infinitely many cases at once: Tₙ = 3n + 1 describes the 1st, the 10th, the 1,000th and the millionth picture of matchstick squares.",{"id":61,"type":62,"items":63},"formulas-nth","formulas",[64,67,70,73,76,79],{"expression":65,"caption":66},"Tₙ = a + (n − 1) × d","Any arithmetic sequence: first term a, common difference d, n − 1 jumps.",{"expression":68,"caption":69},"Tₙ = d × n + (a − d)","The same formula tidied: the step d is the coefficient of n; a − d is the constant.",{"expression":71,"caption":72},"Tₙ = a × rⁿ⁻¹","Any geometric sequence: first term a, common ratio r, n − 1 multiplications.",{"expression":74,"caption":75},"Tₙ = n²","Square numbers.",{"expression":77,"caption":78},"Tₙ = n(n + 1) ÷ 2","Triangular numbers.",{"expression":80,"caption":81},"Fₙ = Fₙ₋₁ + Fₙ₋₂","Fibonacci: each term is built from the two before (a term-to-term rule).",{"id":83,"type":84,"title":85,"problem":86,"steps":87,"help":92},"we-nth","worked_example","Find and check an nth-term formula","Find the nth term of **7, 11, 15, 19, …**, check it, and find the 250th term.",[88,89,90,91],"First term a = 7, common difference d = 4.","Tₙ = a + (n − 1) × d = 7 + (n − 1) × 4 = 7 + 4n − 4 = **4n + 3**.","Check n = 1: 4 + 3 = 7 ✓. n = 2: 8 + 3 = 11 ✓. n = 4: 16 + 3 = 19 ✓.","250th term: 4 × 250 + 3 = **1,003**.",{"simplerExplanation":93},"The sequence goes up in 4s, so it is like the 4 times table (4, 8, 12, 16) but always 3 more. So it is 4n + 3.",{"id":95,"type":96,"caption":97,"columns":98,"rows":104},"table-nth","table","nth-term formulas for some arithmetic sequences (each checked for n = 1 to 20)",[99,100,101,102,103],"Sequence","a","d","Tₙ","T₁₀₀",[105,111,116,122,126,131,137,142],[106,107,108,109,110],"4, 7, 10, 13, …","4","3","3 × n + 1","301",[112,108,113,114,115],"3, 5, 7, 9, …","2","2 × n + 1","201",[117,118,119,120,121],"6, 11, 16, 21, …","6","5","5 × n + 1","501",[123,119,119,124,125],"5, 10, 15, 20, …","5 × n","500",[127,113,128,129,130],"2, 9, 16, 23, …","7","7 × n − 5","695",[132,133,134,135,136],"50, 47, 44, 41, …","50","−3","−3 × n + 53","−247",[138,139,118,140,141],"−4, 2, 8, 14, …","−4","6 × n − 10","590",[143,144,145,146,147],"1.5, 2, 2.5, 3, …","1.5","0.5","0.5 × n + 1","51",{"id":149,"type":47,"variant":150,"title":151,"markdown":152},"mis-3n","misconception","“3n means 3 and n side by side, like 35”","In algebra **3n** always means **3 × n**. If n = 5, then 3n = 15, not 35. Writing 3n instead of 3 × n is just a short cut, and it only works with letters: “3 5” never means 3 × 5. When in doubt, write the × sign; this topic mostly keeps it.",{"id":154,"type":53,"title":155,"eyebrow":156,"navLabel":157},"ch02","Why the step becomes the multiplier","Chapter 02","2 Why it works",{"id":159,"type":43,"markdown":160},"why-step","Here is the reason, once and for all, why an arithmetic sequence with step d always has a formula starting **d × n**.\n\nImagine the **d times table**: d, 2d, 3d, 4d, … Its nth term is d × n, and it goes up by d each time.\n\nYour sequence also goes up by d each time. So your sequence and the d times table **move in step**: whatever the gap between them is at position 1, it stays exactly the same at every position. That gap is a − d (your first term minus the first term of the times table).\n\nSo Tₙ = (d times table) + (fixed gap) = **d × n + (a − d)**.\n\nFor the matchstick squares, the 3 times table is 3, 6, 9, 12 and the sticks are 4, 7, 10, 13: always 1 more. So the rule is 3n + 1. For 50, 47, 44, 41, … (step −3), compare with −3, −6, −9, −12: the gap is always 53, so Tₙ = −3n + 53, better written **53 − 3n**.",{"id":162,"type":84,"title":163,"problem":164,"steps":165},"we-in-sequence","Is a number in the sequence?","Is **500** a term of 7, 11, 15, 19, …? Is **503**? What about **−1** in 50, 47, 44, …?",[166,167,168,169],"For 7, 11, 15, …, Tₙ = 4n + 3. Every term is 3 more than a multiple of 4, so it leaves **remainder 3** when divided by 4.","500 ÷ 4 = 125 remainder 0. Wrong remainder, so **500 is not a term**.","503 ÷ 4 = 125 remainder 3. ✓ Solve 4n + 3 = 503: 4n = 500, **n = 125**, so 503 is the 125th term.","For 50, 47, 44, …, Tₙ = 53 − 3n. Solve 53 − 3n = −1: 3n = 54, **n = 18**. So −1 is the 18th term, and it is the first negative term (the 17th is 2).",{"id":171,"type":172,"component":173,"componentVersion":5,"config":174,"objective":214,"textAlternative":215,"help":216},"lab-nth","interactive","pattern-machine",{"puzzles":175},[176,183,189,193,199,201,207,210],{"kind":177,"rule":178,"show":181,"ask":5,"hint":182},"number",{"type":179,"start":180,"step":181},"add",7,4,"Compare with the 4 times table.",{"kind":177,"rule":184,"show":181,"ask":187,"hint":188},{"type":179,"start":185,"step":186},50,-3,2,"Compare with −3, −6, −9, …",{"kind":177,"rule":190,"show":181,"ask":187},{"type":179,"start":191,"step":192},-4,6,{"kind":177,"rule":194,"show":197,"ask":187,"hint":198},{"type":179,"start":195,"step":196},1.5,0.5,5,"Half steps: the coefficient of n is 0.5.",{"kind":177,"rule":200,"show":181,"ask":187},{"type":179,"start":187,"step":180},{"kind":177,"rule":202,"show":181,"ask":205,"hint":206},{"type":179,"start":203,"step":204},100,-12,3,"Watch for the terms crossing zero.",{"kind":177,"rule":208,"show":181,"ask":5},{"type":179,"start":209,"step":209},11,{"kind":177,"rule":211,"show":181,"ask":187},{"type":179,"start":212,"step":213},-10,-5,"Find the nth-term formula of arithmetic sequences, including decreasing, negative and fractional ones.","Eight arithmetic sequences. After predicting the next terms, you choose the nth-term formula from four options; the wrong options are the classic slips (using the first term as the constant, or the wrong coefficient).\n\n1. 7, 11, 15, 19, … next **23**; Tₙ = **4 × n + 3**.\n2. 50, 47, 44, 41, … next **38, 35**; Tₙ = **−3 × n + 53**.\n3. −4, 2, 8, 14, … next **20, 26**; Tₙ = **6 × n − 10**.\n4. 1.5, 2, 2.5, 3, … next **3.5, 4**; Tₙ = **0.5 × n + 1**.\n5. 2, 9, 16, 23, … next **30, 37**; Tₙ = **7 × n − 5**.\n6. 100, 88, 76, 64, … next **52, 40, 28**; Tₙ = **−12 × n + 112**.\n7. 11, 22, 33, 44, … next **55**; Tₙ = **11 × n**.\n8. −10, −15, −20, −25, … next **−30, −35**; Tₙ = **−5 × n − 5**.\n\nMethod: the step is the coefficient of n. Then choose the constant so that n = 1 gives the first term.",{"hints":217},[218,219],"The constant is the first term minus the step.","Check your formula with n = 2 as well as n = 1.",{"id":221,"type":53,"title":222,"eyebrow":223,"navLabel":224},"ch03","Same pattern, different pictures","Chapter 03","3 Many pictures",{"id":226,"type":43,"markdown":227},"pictures","Different people **see** the same matchstick pattern in different ways, and each way gives a different-looking formula. Here are three ways to count n squares in a row:\n\n| How you see it | Counting | Formula |\n| --- | --- | --- |\n| One stick, then 3 more per square | 1 + 3 + 3 + … | 1 + 3n |\n| First square whole (4), then 3 per extra square | 4 + 3 + 3 + … | 4 + 3(n − 1) |\n| Horizontal sticks and vertical sticks | n on top, n on the bottom, n + 1 upright | 2n + (n + 1) |\n| All squares separate, minus the shared sides | 4 per square, minus n − 1 shared | 4n − (n − 1) |\n\nAll four formulas must give the same numbers, because they count the same sticks. And algebra agrees:\n\n- 4 + 3(n − 1) = 4 + 3n − 3 = 3n + 1\n- 2n + (n + 1) = 3n + 1\n- 4n − (n − 1) = 4n − n + 1 = 3n + 1\n\nExpressions that give the same value for every n are called **equivalent expressions**. Showing that two different pictures lead to equivalent expressions is a small **proof** that both ways of seeing are right.",{"id":229,"type":84,"title":230,"problem":231,"steps":232},"we-hex-pictures","Two ways to see the hexagon chain","Show that “6 sticks for the first hexagon, then 5 for each one after” and “6 per hexagon minus the shared sides” both give 5n + 1.",[233,234,235],"First way: 6 + 5(n − 1) = 6 + 5n − 5 = **5n + 1**.","Second way: n separate hexagons need 6n sticks. In a row of n there are n − 1 shared sides, each counted twice, so subtract n − 1: 6n − (n − 1) = 6n − n + 1 = **5n + 1**.","Both agree. Check n = 3: 6 + 10 = 16 and 18 − 2 = 16 ✓.",{"id":237,"type":172,"component":238,"componentVersion":5,"config":239,"objective":264,"textAlternative":265},"lab-equiv","match-pairs",{"prompt":240,"mode":241,"pairs":242},"Match each way of counting to its simplest equivalent expression.","connect",[243,245,247,249,252,255,258,261],{"a":244,"b":109},"4 + 3 × (n − 1)",{"a":246,"b":114},"3 + 2 × (n − 1)",{"a":248,"b":120},"6 × n − (n − 1)",{"a":250,"b":251},"n + (n − 1)","2 × n − 1",{"a":253,"b":254},"7 + 4 × (n − 1)","4 × n + 3",{"a":256,"b":257},"50 − 3 × (n − 1)","53 − 3 × n",{"a":259,"b":260},"n(n + 1) ÷ 2 + (n + 1)(n + 2) ÷ 2","(n + 1) × (n + 1)",{"a":262,"b":263},"n × n − (n − 1) × (n − 1)","2 × n − 1 (again, from squares)","Show that different ways of counting a pattern lead to equivalent expressions.","Eight pairs of equivalent expressions to connect.\n\n- 4 + 3(n − 1) ↔ 3n + 1 (matchstick squares: first square, then 3 per extra square).\n- 3 + 2(n − 1) ↔ 2n + 1 (matchstick triangles).\n- 6n − (n − 1) ↔ 5n + 1 (hexagons: separate hexagons minus shared sides).\n- n + (n − 1) ↔ 2n − 1 (an L-shape: one arm of n tiles and one of n − 1).\n- 7 + 4(n − 1) ↔ 4n + 3.\n- 50 − 3(n − 1) ↔ 53 − 3n (a decreasing sequence).\n- n(n + 1) ÷ 2 + (n + 1)(n + 2) ÷ 2 ↔ (n + 1)² (two neighbouring triangular numbers make a square).\n- n² − (n − 1)² ↔ 2n − 1 (the gap between neighbouring squares).\n\nTo check a pair, put n = 1, 2 and 3 into both sides, then expand the brackets to be sure.",{"id":267,"type":47,"variant":268,"title":269,"markdown":270},"nuance-three-checks","nuance","Why three checks are not a proof, but expanding is","Two **linear** expressions (like 3n + 1) that agree at two different values of n must agree everywhere, because a straight-line pattern is fixed by any two of its points. But expressions with n² can agree at two values and still differ. Expanding and simplifying both sides is the reliable test, because it shows they are the *same* expression written differently.",{"id":272,"type":53,"title":273,"eyebrow":274,"navLabel":275},"ch04","Powers and geometric growth","Chapter 04","4 Powers",{"id":277,"type":43,"markdown":278},"powers","A geometric sequence multiplies by the same number each time, so its terms are **powers**. We write 2 × 2 × 2 × 2 × 2 as **2⁵** (“2 to the power 5”); the small 5 is the **index** or **exponent** and counts how many 2s are multiplied.\n\n- Powers of 2: 2⁰ = 1, 2¹ = 2, 2² = 4, … 2¹⁰ = **1,024** (about a thousand; that is why computers say “kilo” for 1,024 bytes).\n- Powers of 10: 10⁰ = 1, 10¹ = 10, 10² = 100, 10³ = 1,000, … 10⁵ = 1,00,000 (one lakh), 10⁷ = 1,00,00,000 (one crore).\n- Powers of 3: 1, 3, 9, 27, 81, 243, 729, …\n\nFor a geometric sequence starting at a with ratio r, the first term has had **no** multiplications, the second one, the third two, so the nth term is **a × rⁿ⁻¹**. That is why 3, 6, 12, 24, … has 10th term 3 × 2⁹ = 3 × 512 = 1,536.\n\nBecause 2¹⁰ ≈ 1,000, powers of 2 grow by roughly a factor of 1,000 every 10 steps: 2²⁰ ≈ 10 lakh (actually 1,048,576), and 2³⁰ ≈ 100 crore (actually 1,073,741,824).",{"id":280,"type":84,"title":281,"problem":282,"steps":283},"we-power-sum","The doubling-sum trick","Find **1 + 2 + 4 + 8 + … + 512** without adding term by term, and explain why the sum of powers of 2 is always one less than the next power.",[284,285,286,287,288,289],"Call the sum S. So S = 1 + 2 + 4 + … + 256 + 512.","Double it: 2S = 2 + 4 + 8 + … + 512 + 1,024.","Subtract the first line from the second. Almost everything cancels: 2S − S = 1,024 − 1.","So S = **1,023**.","In general 1 + 2 + 4 + … + 2ⁿ⁻¹ = **2ⁿ − 1**. That is why Bela’s totals in Investigate were 1, 3, 7, 15, 31, …","A picture: if you have coins worth ₹1, 2, 4, …, 512, the next coin (₹1,024) is worth exactly ₹1 more than all the others together.",{"id":291,"type":47,"variant":292,"title":293,"markdown":294},"aha-binary","aha","Every whole number is a sum of different powers of 2","Because each power of 2 is one more than all the smaller ones added together, you can make **every** whole number from 1 to 1,023 using each of 1, 2, 4, …, 512 at most once. For example 100 = 64 + 32 + 4. This is the idea behind **binary**, the number system computers use (100 in binary is 1100100). You will explore it in the Extend layer.",{"id":296,"type":53,"title":297,"eyebrow":298,"navLabel":299},"ch05","Squares, triangles and sums: proofs with pictures","Chapter 05","5 Squares & sums",{"id":301,"type":43,"markdown":302},"second-diff","For square numbers the **first differences** are not constant, but the **second differences** (the differences of the differences) are:\n\n| n | 1 | 2 | 3 | 4 | 5 | 6 |\n| --- | --- | --- | --- | --- | --- | --- |\n| n² | 1 | 4 | 9 | 16 | 25 | 36 |\n| 1st difference | | 3 | 5 | 7 | 9 | 11 |\n| 2nd difference | | | 2 | 2 | 2 | 2 |\n\nA constant **first** difference means a **linear** formula (like 3n + 1). A constant **second** difference means the formula has an **n²** in it: it is **quadratic**. For triangular numbers the second difference is 1, and indeed n(n + 1) ÷ 2 = (n² + n) ÷ 2 has n² in it. A constant **ratio** means an **exponential** (geometric) formula like 2ⁿ.\n\nThis gives you a checklist for any sequence: constant 1st difference → linear; constant 2nd difference → quadratic; constant ratio → exponential; each term the sum of the two before → Fibonacci-type.",{"id":304,"type":305,"title":306,"items":307},"steps-proof-tri","steps","Proof: 1 + 2 + … + n = n(n + 1) ÷ 2",[308,311,314,317,320,323],{"title":309,"text":310},"Name the sum","S = 1 + 2 + 3 + … + (n − 1) + n.",{"title":312,"text":313},"Write it backwards","S = n + (n − 1) + … + 3 + 2 + 1.",{"title":315,"text":316},"Add the two lines column by column","Each column adds to n + 1: 1 + n, 2 + (n − 1), 3 + (n − 2), …",{"title":318,"text":319},"Count the columns","There are n columns, so 2S = n × (n + 1).",{"title":321,"text":322},"Halve","S = n(n + 1) ÷ 2. This is the two-staircase rectangle from Investigate, written in algebra.",{"title":324,"text":325},"Check","n = 100: 100 × 101 ÷ 2 = 5,050 ✓ (Gauss). n = 4: 4 × 5 ÷ 2 = 10 ✓.",{"id":327,"type":305,"title":328,"items":329},"steps-proof-odd","Proof: 1 + 3 + 5 + … + (2n − 1) = n²",[330,333,336,339,342],{"title":331,"text":332},"The nth odd number","The odd numbers 1, 3, 5, … have step 2 and first term 1, so the nth is 2n − 1.",{"title":334,"text":335},"Picture","An L-shape of 2n − 1 tiles wraps an (n − 1) square into an n square.",{"title":337,"text":338},"Stack the Ls","Start with 1 tile, wrap L-shapes of 3, 5, 7, …, 2n − 1. You build an n by n square.",{"title":340,"text":341},"Algebra","Pairing works too: (1 + (2n − 1)) × n ÷ 2 = 2n × n ÷ 2 = n².",{"title":324,"text":343},"n = 10: 1 + 3 + … + 19 = 100 ✓.",{"id":345,"type":47,"variant":292,"title":346,"markdown":347},"aha-cubes","A beautiful surprise: sums of cubes","Add the cube numbers: 1 = 1, 1 + 8 = 9, 1 + 8 + 27 = 36, 1 + 8 + 27 + 64 = 100.\n\nThese are 1², 3², 6², 10²: the **squares of the triangular numbers**! So 1³ + 2³ + … + n³ = (1 + 2 + … + n)². For n = 10 that gives 55² = **3,025**. The Indian mathematician **Āryabhaṭa** gave rules for both the sum of squares and the sum of cubes in his *Āryabhaṭīya* (499 CE).",{"id":349,"type":84,"title":350,"problem":351,"steps":352,"help":357},"we-series","Adding any arithmetic sequence","A savings plan puts in ₹100 in month 1, ₹120 in month 2, ₹140 in month 3, and so on, rising by ₹20 each month. How much has been saved after **2 years** (24 months)?",[353,354,355,356],"The monthly amounts are arithmetic: a = 100, d = 20.","Month 24: Tₙ = 100 + 23 × 20 = **₹560**.","Sum = (first + last) × number of terms ÷ 2 (Gauss's pairing).","Sum = (100 + 560) × 24 ÷ 2 = 660 × 12 = **₹7,920**.",{"simplerExplanation":358},"Pair month 1 with month 24 (₹100 + ₹560 = ₹660), month 2 with month 23, and so on. There are 12 pairs of ₹660.",{"id":360,"type":361,"itemId":362,"prompt":363,"check":364,"hints":367,"feedback":370},"pr-sum-cubes","practice","patterns.deepen-sum-cubes","Use the pattern for sums of cubes to find **1³ + 2³ + 3³ + … + 6³**.",{"kind":177,"answer":365,"tolerance":366},441,0,[368,369],"First find 1 + 2 + … + 6.","Then square it.",{"correct":371,"incorrect":372},"Yes: 1 + 2 + … + 6 = 21, and 21² = 441.","The sum of the first n cubes is (1 + 2 + … + n)². Here 1 + … + 6 = 21, and 21 × 21 = 441. Check: 1 + 8 + 27 + 64 + 125 + 216 = 441.",{"id":374,"type":172,"component":375,"componentVersion":5,"config":376,"objective":456,"textAlternative":457,"help":458},"lab-sort-growth","sort-game",{"prompt":377,"bins":378,"items":391,"seconds":366},"Use differences and ratios to sort each sequence by the kind of formula that makes it.",[379,382,385,388],{"id":380,"label":381},"linear","Linear (1st difference constant)",{"id":383,"label":384},"quadratic","Quadratic (2nd difference constant)",{"id":386,"label":387},"exponential","Exponential (ratio constant)",{"id":389,"label":390},"fib","Fibonacci-type",[392,396,400,404,408,412,416,420,424,428,432,436,440,444,448,452],{"id":393,"label":394,"bin":380,"why":395},"q1","5, 8, 11, 14, 17","First differences all 3: Tₙ = 3n + 2.",{"id":397,"label":398,"bin":383,"why":399},"q2","2, 5, 10, 17, 26","Differences 3, 5, 7, 9; second differences 2: Tₙ = n² + 1.",{"id":401,"label":402,"bin":386,"why":403},"q3","3, 6, 12, 24, 48","Ratio 2 each time: Tₙ = 3 × 2ⁿ⁻¹.",{"id":405,"label":406,"bin":389,"why":407},"q4","2, 1, 3, 4, 7, 11","Each term is the sum of the two before (the Lucas numbers).",{"id":409,"label":410,"bin":383,"why":411},"q5","1, 3, 6, 10, 15","Differences 2, 3, 4, 5; second differences 1: n(n + 1) ÷ 2.",{"id":413,"label":414,"bin":380,"why":415},"q6","100, 91, 82, 73","First differences all −9: Tₙ = 109 − 9n.",{"id":417,"label":418,"bin":386,"why":419},"q7","1, 10, 100, 1,000","Ratio 10: powers of 10.",{"id":421,"label":422,"bin":383,"why":423},"q8","2, 6, 12, 20, 30","Second differences 2: n(n + 1).",{"id":425,"label":426,"bin":389,"why":427},"q9","5, 5, 10, 15, 25, 40","Fibonacci rule from 5, 5: every term is 5 × a Fibonacci number.",{"id":429,"label":430,"bin":386,"why":431},"q10","64, 32, 16, 8, 4","Ratio ½ each time.",{"id":433,"label":434,"bin":383,"why":435},"q11","0, 3, 8, 15, 24","Second differences 2: n² − 1.",{"id":437,"label":438,"bin":380,"why":439},"q12","−7, −2, 3, 8, 13","First differences all 5: Tₙ = 5n − 12.",{"id":441,"label":442,"bin":389,"why":443},"q13","1, 3, 4, 7, 11, 18","Each term is the sum of the two before, starting 1, 3.",{"id":445,"label":446,"bin":386,"why":447},"q14","1, 3, 9, 27, 81","Ratio 3: powers of 3.",{"id":449,"label":450,"bin":383,"why":451},"q15","4, 7, 12, 19, 28","Differences 3, 5, 7, 9; second differences 2: n² + 3.",{"id":453,"label":454,"bin":380,"why":455},"q16","0.5, 1, 1.5, 2, 2.5","First differences all 0.5: Tₙ = 0.5n.","Classify sequences as linear, quadratic, exponential or Fibonacci-type by testing differences and ratios.","Sixteen sequences, four bins.\n\n**Linear** (constant first difference): 5, 8, 11, 14, 17 (3n + 2); 100, 91, 82, 73 (109 − 9n); −7, −2, 3, 8, 13 (5n − 12); 0.5, 1, 1.5, 2, 2.5 (0.5n).\n\n**Quadratic** (constant second difference): 2, 5, 10, 17, 26 (n² + 1); 1, 3, 6, 10, 15 (triangular); 2, 6, 12, 20, 30 (n(n + 1)); 0, 3, 8, 15, 24 (n² − 1); 4, 7, 12, 19, 28 (n² + 3).\n\n**Exponential** (constant ratio): 3, 6, 12, 24, 48; 1, 10, 100, 1,000; 64, 32, 16, 8, 4; 1, 3, 9, 27, 81.\n\n**Fibonacci-type** (sum of the two before): 2, 1, 3, 4, 7, 11 (Lucas); 5, 5, 10, 15, 25, 40; 1, 3, 4, 7, 11, 18.",{"hints":459},[460,461],"Work out first differences. If they are not constant, work out second differences.","If the differences look like the sequence itself, try ratios.",{"id":463,"type":53,"title":464,"eyebrow":465,"navLabel":466},"ch06","Fibonacci numbers: an Indian discovery","Chapter 06","6 Fibonacci in India",{"id":468,"type":43,"markdown":469},"prosody","Sanskrit and Prakrit poetry is built from syllables that are **short** (*laghu*, 1 beat) or **long** (*guru*, 2 beats). Poets and scholars of **prosody** (*chandaḥśāstra*, the science of metre) asked a natural question: **how many different rhythms fill a line of exactly n beats?**\n\nTry it with S for a short syllable (1 beat) and L for a long one (2 beats):\n\n| Beats | Rhythms | How many |\n| --- | --- | --- |\n| 1 | S | 1 |\n| 2 | SS, L | 2 |\n| 3 | SSS, SL, LS | 3 |\n| 4 | SSSS, SSL, SLS, LSS, LL | 5 |\n| 5 | SSSSS, SSSL, SSLS, SLSS, LSSS, SLL, LSL, LLS | 8 |\n\nThe counts go **1, 2, 3, 5, 8, 13, 21, 34**, …: the Fibonacci numbers! **Why?** Every rhythm of n beats ends in either a short syllable or a long one. If it ends in S, the rest is a rhythm of n − 1 beats; if it ends in L, the rest is a rhythm of n − 2 beats. So (rhythms of n) = (rhythms of n − 1) + (rhythms of n − 2). That is exactly the Fibonacci rule, and it is a real **proof**, not just a spotted pattern.",{"id":471,"type":472,"title":473,"items":474},"timeline-history","timeline","Who found these patterns?",[475,479,483,487,491,495,499,503,507,511,515],{"time":476,"title":477,"text":478},"3rd–2nd BCE","Pingala","Pingala’s *Chandaḥśāstra*, a treatise on metre, studies patterns of short and long syllables. Its rules contain the seeds of both the Fibonacci numbers and the triangle later called Meru Prastara — though the Fibonacci rule itself is only stated plainly by later writers.",{"time":480,"title":481,"text":482},"499 CE","Āryabhaṭa","The *Āryabhaṭīya* gives rules for sums of arithmetic sequences and for sums of squares and cubes.",{"time":484,"title":485,"text":486},"c. 600–800","Virahānka","Gives the rule clearly: the number of rhythms of n beats is the sum of the counts for n − 1 and n − 2. NCERT calls these Virahānka numbers.",{"time":488,"title":489,"text":490},"10th century","Halāyudha","His commentary on Pingala describes the Meru Prastara (“staircase of Mount Meru”): the triangle of numbers now called Pascal’s triangle.",{"time":492,"title":493,"text":494},"12th century","Khajuraho","A 4 × 4 magic square is inscribed on the wall of the Parshvanath temple, which was itself built about 950–970 CE (see Extend).",{"time":496,"title":497,"text":498},"c. 1135","Gopāla","Also gives the rhythm-counting rule for these numbers.",{"time":500,"title":501,"text":502},"c. 1150","Hemachandra","The Jain scholar Hemachandra states the same rule for counting metres, still about half a century before Fibonacci.",{"time":504,"title":505,"text":506},"1202","Fibonacci","Leonardo of Pisa’s *Liber Abaci* brings Hindu–Arabic numerals to Europe and includes the rabbit puzzle that gives 1, 1, 2, 3, 5, 8, …",{"time":508,"title":509,"text":510},"1356","Nārāyaṇa Paṇḍita","His *Gaṇita Kaumudī* studies these sequences further, and magic squares in great detail.",{"time":512,"title":513,"text":514},"1654","Pascal","Blaise Pascal writes his treatise on the arithmetic triangle; in Europe it takes his name.",{"time":516,"title":517,"text":518},"1870s","Lucas","Édouard Lucas attaches Fibonacci’s name to the sequence and studies its relative 2, 1, 3, 4, 7, 11, …",{"id":520,"type":47,"variant":268,"title":521,"markdown":522},"nuance-names","Names in mathematics are often unfair","Mathematical ideas are frequently named after the person who made them famous in Europe, not the first person to find them. The Fibonacci numbers were known in India for centuries before 1202, and the “Pascal” triangle was known in India, Persia (al-Karajī, Omar Khayyām) and China (Jia Xian, Yang Hui) long before 1654. It is fine to use the common names, as long as you know the fuller story.",{"id":524,"type":96,"caption":525,"columns":526,"rows":529},"table-ratio","Divide each Fibonacci number by the one before (to 4 decimal places)",[527,528],"Fraction","Decimal",[530,533,536,539,542,545,548,551,554,557],[531,532],"2 ÷ 1","2.0000",[534,535],"3 ÷ 2","1.5000",[537,538],"5 ÷ 3","1.6667",[540,541],"8 ÷ 5","1.6000",[543,544],"13 ÷ 8","1.6250",[546,547],"21 ÷ 13","1.6154",[549,550],"34 ÷ 21","1.6190",[552,553],"55 ÷ 34","1.6176",[555,556],"89 ÷ 55","1.6182",[558,559],"144 ÷ 89","1.6180",{"id":561,"type":43,"markdown":562},"golden","The ratios swing above and below a special number, getting closer each time: **1.6180339…**, called the **golden ratio** and written φ (the Greek letter *phi*). It equals (1 + √5) ÷ 2.\n\nThis is the usual explanation for why Fibonacci numbers show up in plants. A growing sunflower adds each new seed at roughly a fixed angle of turn from the last, about 137.5°, which is 360° ÷ φ² (the “golden angle”). That angle packs the seeds evenly with no wasted gaps, and the spirals your eye picks out then come in neighbouring Fibonacci numbers: often **34 and 55**, or **55 and 89** in large heads. Pinecones commonly show **8 and 13** spirals and pineapples **8, 13 and 21**.\n\nIt is a strong tendency, not a law. In a citizen-science count of 657 sunflower heads, 565 of 768 careful spiral counts were Fibonacci numbers, but 136 of them — nearly one in five — had no Fibonacci structure at all. Some plants show neighbouring Lucas numbers (like 4, 7, 11) instead. So: common, and for a reason, but always worth counting rather than assuming.",{"id":564,"type":361,"itemId":565,"prompt":566,"check":567,"hints":569,"feedback":572},"pr-rhythm","patterns.deepen-rhythms","How many different rhythms, using short (1 beat) and long (2 beat) syllables, fill a line of exactly **8 beats**?",{"kind":177,"answer":568,"tolerance":366},34,[570,571],"The counts for 1, 2, 3, 4, 5 beats are 1, 2, 3, 5, 8.","Keep adding the last two counts.",{"correct":573,"incorrect":574},"Yes: 1, 2, 3, 5, 8, 13, 21, 34. There are 34 rhythms of 8 beats.","Each count is the sum of the two before: 6 beats → 13, 7 beats → 21, 8 beats → 13 + 21 = 34.",{"id":576,"type":172,"component":173,"componentVersion":5,"config":577,"objective":611,"textAlternative":612},"lab-fib",{"puzzles":578},[579,583,586,588,590,602],{"kind":177,"rule":580,"show":197,"ask":205,"hint":582},{"type":581,"a":5,"b":187},"fibonacci","Rhythms of 1, 2, 3, 4, 5 beats.",{"kind":177,"rule":584,"show":192,"ask":187,"hint":585},{"type":581,"a":187,"b":5},"Lucas numbers: Fibonacci’s rule, different start.",{"kind":177,"rule":587,"show":197,"ask":187},{"type":581,"a":205,"b":205},{"kind":177,"rule":589,"show":197,"ask":187},{"type":581,"a":5,"b":181},{"kind":177,"rule":591,"show":192,"ask":187,"hint":601},{"type":592,"terms":593,"ruleText":600},"list",[5,187,181,180,594,595,596,597,598,599],12,20,33,54,88,143,"Running totals of Fibonacci numbers: each is one less than a Fibonacci number","Add 1 to each term.",{"kind":177,"rule":603,"show":181,"ask":5,"hint":610},{"type":592,"terms":604,"ruleText":609},[187,605,568,606,607,608],8,144,610,2584,"Every third Fibonacci number: the even ones (each is 4 × the last + the one before)","Which Fibonacci numbers are even?","Extend Fibonacci-type sequences from different starts and spot hidden Fibonacci patterns.","Six Fibonacci puzzles.\n\n1. Pingala–Virahānka rhythm counts: 1, 2, 3, 5, 8, … next **13, 21, 34**.\n2. Lucas numbers: 2, 1, 3, 4, 7, 11, … next **18, 29**.\n3. Starting 3, 3: 3, 3, 6, 9, 15, … next **24, 39** (3 × the Fibonacci numbers).\n4. Starting 1, 4: 1, 4, 5, 9, 14, … next **23, 37**.\n5. Running totals of 1, 1, 2, 3, 5, 8, …: 1, 2, 4, 7, 12, 20, … next **33, 54**. Each total is one less than a Fibonacci number (1 + 1 + 2 + 3 + 5 + 8 = 20 = 21 − 1).\n6. The even Fibonacci numbers: 2, 8, 34, 144, … next **610**. They are every third Fibonacci number.",{"id":614,"type":53,"title":615,"eyebrow":616,"navLabel":617},"ch07","Meru Prastara: the triangle of patterns","Chapter 07","7 Meru Prastara",{"id":619,"type":43,"markdown":620},"meru","Write a 1 at the top. Each row below starts and ends with 1, and every other number is the **sum of the two numbers just above it**. Halāyudha, explaining Pingala’s rules in the 10th century, called this the **Meru Prastara**, the “staircase of Mount Meru”. In Europe it is called **Pascal’s triangle**.\n\nPingala’s question was: *in a line of n syllables, in how many ways can exactly k of them be long?* The answer is the kth number in row n of the triangle (counting from 0). For example, in 4 syllables there are 6 ways to have exactly 2 long ones: LLSS, LSLS, LSSL, SLLS, SLSL, SSLL.",{"id":622,"type":96,"caption":623,"columns":624,"rows":628},"table-meru","Meru Prastara, rows 0 to 8 (each entry the sum of the two above)",[625,626,627],"Row","Numbers","Row total",[629,632,634,636,639,642,645,648,651],[630,631,631],"0","1",[631,633,113],"1  1",[113,635,107],"1  2  1",[108,637,638],"1  3  3  1","8",[107,640,641],"1  4  6  4  1","16",[119,643,644],"1  5  10  10  5  1","32",[118,646,647],"1  6  15  20  15  6  1","64",[128,649,650],"1  7  21  35  35  21  7  1","128",[638,652,653],"1  8  28  56  70  56  28  8  1","256",{"id":655,"type":656,"title":657,"prompt":658,"options":659},"explorer-meru","explorer","Patterns hiding in the Meru Prastara","Pick a pattern to see where it hides in the triangle.",[660,666,672,679,687,692],{"id":661,"label":662,"chain":663,"note":665},"rowsum","Row totals",[631,113,107,638,641,664],"Powers of 2","Each row adds up to double the row before, because every number is used twice in making the next row. Row n totals 2ⁿ. This matches Pingala’s count: n syllables, each short or long, give 2ⁿ patterns altogether.",{"id":667,"label":668,"chain":669,"note":671},"diag1","Second diagonal",[631,113,108,107,119,670],"Counting numbers","The numbers just inside the edge are the counting numbers 1, 2, 3, 4, … (the number of ways to choose 1 long syllable from n).",{"id":673,"label":674,"chain":675,"note":678},"diag2","Third diagonal",[631,108,118,33,676,677],"15","Triangular numbers","The next diagonal holds the triangular numbers 1, 3, 6, 10, 15, 21, 28, … (the number of ways to choose 2 things from n + 1, which is also the number of handshakes among n + 1 people).",{"id":680,"label":681,"chain":682,"note":686},"diag3","Fourth diagonal",[631,107,33,683,684,685],"20","35","Tetrahedral numbers","Next come 1, 4, 10, 20, 35, …, the tetrahedral numbers: the number of balls in a triangular pyramid, like fruit stacked at a market. Each is a running total of the triangular numbers.",{"id":389,"label":688,"chain":689,"note":691},"Shallow diagonals",[631,631,113,108,119,690],"Fibonacci!","Add the numbers along the shallow slanting lines (up one row and across one) and you get 1, 1, 2, 3, 5, 8, 13, … The two Pingala questions, rhythms of n beats and choices of long syllables, are linked through the triangle.",{"id":693,"label":694,"chain":695,"note":701},"eleven","Powers of 11",[631,696,697,698,699,700],"11","121","1331","14641","Then carrying","Rows 0 to 4 read as numbers are 11⁰ to 11⁴. Row 5 (1 5 10 10 5 1) would need carrying: 11⁵ = 161,051. Another pattern that breaks when a “digit” reaches 10!",{"id":703,"type":361,"itemId":704,"prompt":705,"check":706,"hints":708,"feedback":711},"pr-meru","patterns.deepen-meru-row","Row 8 of the Meru Prastara is 1 8 28 56 70 56 28 8 1. What does **row 9** add up to, and what is the middle pair of numbers in row 9? Enter the **row total**.",{"kind":177,"answer":707,"tolerance":366},512,[709,710],"Each row total is double the one before.","Row 8 totals 256.",{"correct":712,"incorrect":713},"Yes: 512. Row 9 is 1 9 36 84 126 126 84 36 9 1, with 126 and 126 in the middle.","Row totals double each time: row 8 totals 256, so row 9 totals 512. (Row 9 is 1 9 36 84 126 126 84 36 9 1.)",{"id":715,"type":53,"title":716,"eyebrow":717,"navLabel":718},"ch08","Why the digit patterns work, and why they stop","Chapter 08","8 Digit proofs",{"id":720,"type":43,"markdown":721},"digit-proof","**Why do the digits of multiples of 9 add to 9?** Take a two-digit number with tens digit a and units digit b. Its value is 10a + b = **9a + (a + b)**. So the number and its digit sum a + b differ by 9a, a multiple of 9. If the number is a multiple of 9, its digit sum must be a multiple of 9 too. For 9 × 1 to 9 × 10 the digit sum is exactly 9. (For bigger multiples, like 99 = 9 × 11, the digit sum is 18, and adding those digits gives 9.) This is the reason behind the **divisibility test for 9**.\n\n**Why does 111 × 111 = 12,321?** Multiply the long way. Each row of the working is 111, shifted one place:\n\n| | | | | | |\n| --- | --- | --- | --- | --- | --- |\n| | | | 1 | 1 | 1 |\n| | | 1 | 1 | 1 | |\n| | 1 | 1 | 1 | | |\n| Column sums | 1 | 2 | 3 | 2 | 1 |\n\nThe number of 1s in each column rises and falls, giving 1, 2, 3, 2, 1. With nine 1s the middle column holds 9, the biggest single digit. With ten 1s the middle column would hold 10, which forces a carry and the pattern breaks. You predicted this in Understand; now you know **why**.",{"id":723,"type":84,"title":724,"problem":725,"steps":726},"we-9-trick","Why 12 × 9 + 3 = 111","Explain the pattern 1 × 9 + 2 = 11, 12 × 9 + 3 = 111, 123 × 9 + 4 = 1,111.",[727,728,729,730],"Multiplying by 9 is multiplying by 10 and subtracting once: 12 × 9 = 120 − 12.","So 12 × 9 + 3 = 120 − 12 + 3 = 111.","For the next line: 123 × 9 + 4 = 1,230 − 123 + 4 = 1,111.","In general, if the number is 123…k, then 10 × (123…k) − (123…k) lines up digits so that each column gives 1 after borrowing, and the + (k + 1) fixes the last column. The pattern needs every digit 1 to 9 to be a single digit; it runs to 123,456,789 × 9 + 10 = 1,111,111,111 and then stops.",{"id":732,"type":53,"title":733,"eyebrow":734,"navLabel":735},"ch09","Patterns that break: why proof matters","Chapter 09","9 Patterns that break",{"id":737,"type":43,"markdown":738},"circle","Here is the most famous trap in the study of patterns. Draw a circle and mark some points on it. Join every pair of points with a straight line, and count the regions (pieces) the circle is cut into. Place the points so that no three lines meet at the same point inside, which gives the most regions.\n\n| Points | 1 | 2 | 3 | 4 | 5 | 6 |\n| --- | --- | --- | --- | --- | --- | --- |\n| Regions | 1 | 2 | 4 | 8 | 16 | 31 |\n\n1, 2, 4, 8, 16, … Surely the next is 32? **It is 31.** Draw it very carefully with 6 points and count: 31 is the most you can get, however hard you try. (Watch out when you draw: six *equally spaced* points give only **30**, because the three long diagonals all cross at the centre. Nudge one point and you get 31.) With 7 points you get 57, not 64, and with 10 points just 256, not 512.\n\nThe true rule has nothing to do with doubling. The regions are 1 + (number of lines) + (number of crossing points inside): each line and each crossing adds one region. With n points there are n(n − 1) ÷ 2 lines, and every choice of 4 points gives exactly one crossing. The doubling was a coincidence of small numbers.",{"id":740,"type":741,"prompt":742,"options":743,"explanation":754},"predict-circle","prediction","Before reading the table below, **predict**: with **6 points** on a circle, all joined, into how many regions is the circle cut (at most)?",[744,746,749,752],{"id":100,"label":745},"32, because it doubles",{"id":747,"label":748},"b","31",{"id":750,"label":751},"c","30",{"id":101,"label":753},"36","**31.** The first five answers, 1, 2, 4, 8, 16, trick almost everybody into expecting 32. With 6 points there are 15 lines and 15 crossing points, giving 1 + 15 + 15 = **31** regions. You only get all 31 if no three lines cross at the same point: six equally spaced points give 30, because the three long diagonals meet at the centre. This is sometimes called **Moser’s circle problem**.",{"id":756,"type":172,"component":173,"componentVersion":5,"config":757,"objective":797,"textAlternative":798},"lab-break",{"puzzles":758},[759,770,773,782],{"kind":177,"rule":760,"show":197,"ask":5,"hint":769},{"type":592,"terms":761,"ruleText":768},[5,187,181,605,762,763,764,765,766,767],16,31,57,99,163,256,"Regions of a circle cut by lines joining n points: 1 + lines + crossings (not doubling!)","It looks like doubling. Is it really?",{"kind":177,"rule":771,"show":197,"ask":5},{"type":772,"start":5,"factor":187},"multiply",{"kind":177,"rule":774,"show":205,"ask":205,"hint":781},{"type":592,"terms":775,"ruleText":780},[5,187,181,180,209,762,776,777,778,779],22,29,37,46,"Add 1, then 2, then 3, …: pieces from straight cuts of a pancake (1 + n(n + 1) ÷ 2)","Three terms can fit many rules. Look at the differences.",{"kind":177,"rule":783,"show":192,"ask":187,"hint":796},{"type":592,"terms":784,"ruleText":795},[785,786,787,788,789,790,791,792,793,794],41,43,47,53,61,71,83,97,113,131,"n × n − n + 41: prime for n = 1 to 40, but not for n = 41","Differences 2, 4, 6, 8, …","Discover that patterns which look certain can break, and that different rules can share the same first terms.","Four puzzles about trust.\n\n1. Circle regions: 1, 2, 4, 8, 16, … Most people type 32. The real next term is **31**. The rule is 1 + lines + crossings, not doubling.\n2. Doubling from 1: 1, 2, 4, 8, 16, … next **32**. The same first five terms as puzzle 1, but a different rule.\n3. Pancake cuts (the most pieces from n straight cuts of a round pancake, counting 0 cuts as 1 piece): 1, 2, 4, … next **7, 11, 16** (add 1, 2, 3, 4, 5). The same first three terms as doubling!\n4. n × n − n + 41: 41, 43, 47, 53, 61, 71, … next **83, 97**. Every one of these is prime, all the way to n = 40 (1,601). But n = 41 gives 41 × 41 = 1,681, which is not prime.\n\nMoral: a rule is only safe when you know **why** it works.",{"id":800,"type":47,"variant":292,"title":801,"markdown":802},"aha-proof","This is why mathematicians prove things","Scientists test ideas with experiments, and a theory that passes thousands of tests is trusted. Mathematicians go further: a statement about **all** numbers is accepted only when there is a **proof**, an argument that covers every case at once. The matchstick formula 3n + 1, the odd-number sum n², and Gauss’s n(n + 1) ÷ 2 are proved. The circle “doubling” was only guessed, and it failed at the sixth step. Some guesses in mathematics have been checked by computer for every number up to more than 10¹⁸ (a million million million) and are still unproved (you will meet one in Extend).",{"id":804,"type":47,"variant":805,"title":806,"markdown":807},"careful-sequence-questions","careful","What “next term” questions really mean","In tests, “find the next term” means “find the simplest rule that fits and use it”. Always show the rule you used, so a reader can check your thinking. If you notice that two rules fit, say so: that is a sign of a strong mathematician, not a weak one.",{"id":809,"type":53,"title":810,"eyebrow":811,"navLabel":812},"ch10","Check your deeper understanding","Chapter 10","10 Check yourself",{"id":814,"type":815,"title":816,"terms":817},"glossary-deepen","glossary","Algebra and pattern words",[818,822,826,830,834,838,842,846,850,854,858,862,866,870,874,878,882],{"term":819,"meaning":820,"example":821},"variable","A letter that stands for a number that can change, such as the position n.","In 3n + 1, n is the variable.",{"term":823,"meaning":824,"example":825},"expression","A combination of numbers, letters and operations, without an equals sign.","4n + 3",{"term":827,"meaning":828,"example":829},"formula","An equation that tells you how to calculate one quantity from others.","Tₙ = 4n + 3",{"term":831,"meaning":832,"example":833},"coefficient","The number that multiplies a variable.","In 5n + 1, the coefficient of n is 5.",{"term":835,"meaning":836,"example":837},"constant","A number on its own in an expression, not multiplied by a variable.","In 5n + 1, the constant is 1.",{"term":839,"meaning":840,"example":841},"equivalent expressions","Expressions that give the same value for every value of the variable.","4 + 3(n − 1) and 3n + 1",{"term":843,"meaning":844,"example":845},"linear sequence","A sequence with a constant first difference; its nth term has the form dn + c.","5, 8, 11, 14",{"term":847,"meaning":848,"example":849},"quadratic sequence","A sequence with a constant second difference; its nth term has n² in it.","2, 5, 10, 17 (n² + 1)",{"term":851,"meaning":852,"example":853},"second difference","The difference between neighbouring first differences.","For 1, 4, 9, 16 it is always 2.",{"term":855,"meaning":856,"example":857},"power \u002F index","A number multiplied by itself; the index says how many times.","2⁵ = 32",{"term":859,"meaning":860,"example":861},"exponential growth","Growth by the same multiplying factor each step.","Doubling: 1, 2, 4, 8, 16",{"term":863,"meaning":864,"example":865},"golden ratio","The number (1 + √5) ÷ 2 ≈ 1.618, approached by ratios of neighbouring Fibonacci numbers.","89 ÷ 55 ≈ 1.618",{"term":867,"meaning":868,"example":869},"Meru Prastara","The triangle of numbers (Pascal’s triangle) described in Halāyudha’s commentary on Pingala; each number is the sum of the two above.","Row 4: 1 4 6 4 1",{"term":871,"meaning":872,"example":873},"prosody (chandaḥśāstra)","The study of the rhythm and metre of poetry; Indian prosody led to early discoveries about patterns.","Short and long syllables",{"term":875,"meaning":876,"example":877},"Lucas numbers","The Fibonacci rule started from 2 and 1.","2, 1, 3, 4, 7, 11, 18",{"term":879,"meaning":880,"example":881},"proof","An argument showing a statement is true in every case, not just the ones checked.","Pairing proves 1 + … + n = n(n + 1) ÷ 2.",{"term":883,"meaning":884,"example":885},"tetrahedral number","The number of balls in a triangular pyramid; a running total of triangular numbers.","1, 4, 10, 20, 35",{"id":887,"type":888,"title":889,"questions":890},"quiz-deepen","quiz","Reasoning with patterns",[891,904,917,928,941,951,964,977,989,998,1011],{"itemId":892,"prompt":893,"options":894,"correct":747,"why":903},"patterns.deepen-q-nth","What is the nth term of 9, 14, 19, 24, …?",[895,897,899,901],{"id":100,"label":896},"5n + 9",{"id":747,"label":898},"5n + 4",{"id":750,"label":900},"9n + 5",{"id":101,"label":902},"4n + 5","Step 5 gives 5n. For n = 1 we need 9, so add 4: 5n + 4.",{"itemId":905,"prompt":906,"options":907,"correct":747,"why":916},"patterns.deepen-q-dec","What is the nth term of 40, 36, 32, 28, …?",[908,910,912,914],{"id":100,"label":909},"4n + 36",{"id":747,"label":911},"44 − 4n",{"id":750,"label":913},"40 − 4n",{"id":101,"label":915},"36 − 4n","Step −4 gives −4n. For n = 1 we need 40, so the constant is 44: 44 − 4n.",{"itemId":918,"prompt":919,"options":920,"correct":750,"why":927},"patterns.deepen-q-in","Is 300 a term of 5, 12, 19, 26, … (7n − 2)?",[921,923,925],{"id":100,"label":922},"Yes, the 43rd",{"id":747,"label":924},"Yes, the 42nd",{"id":750,"label":926},"No","7n − 2 = 300 gives 7n = 302, and 302 is not a multiple of 7 (7 × 43 = 301). So 300 is not a term.",{"itemId":929,"prompt":930,"options":931,"correct":747,"why":940},"patterns.deepen-q-equiv","Which expression is equivalent to 6 + 5(n − 1)?",[932,934,936,938],{"id":100,"label":933},"5n + 6",{"id":747,"label":935},"5n + 1",{"id":750,"label":937},"6n − 1",{"id":101,"label":939},"11n − 5","6 + 5n − 5 = 5n + 1 (the hexagon chain).",{"itemId":942,"prompt":943,"options":944,"correct":747,"why":950},"patterns.deepen-q-second","The sequence 3, 6, 11, 18, 27, … has second differences of:",[945,946,947,948],{"id":100,"label":631},{"id":747,"label":113},{"id":750,"label":108},{"id":101,"label":949},"They are not constant","First differences 3, 5, 7, 9; second differences 2. It is n² + 2.",{"itemId":952,"prompt":953,"options":954,"correct":747,"why":963},"patterns.deepen-q-pow","What is 1 + 2 + 4 + … + 2⁹?",[955,957,959,961],{"id":100,"label":956},"512",{"id":747,"label":958},"1,023",{"id":750,"label":960},"1,024",{"id":101,"label":962},"1,022","The sum of powers of 2 up to 2⁹ is 2¹⁰ − 1 = 1,023.",{"itemId":965,"prompt":966,"options":967,"correct":100,"why":976},"patterns.deepen-q-cubes","What is 1³ + 2³ + … + 5³?",[968,970,972,974],{"id":100,"label":969},"225",{"id":747,"label":971},"125",{"id":750,"label":973},"100",{"id":101,"label":975},"215","(1 + 2 + 3 + 4 + 5)² = 15² = 225.",{"itemId":978,"prompt":979,"options":980,"correct":750,"why":988},"patterns.deepen-q-rhythm","How many rhythms of short (1 beat) and long (2 beats) syllables fill 6 beats?",[981,982,984,986],{"id":100,"label":638},{"id":747,"label":983},"12",{"id":750,"label":985},"13",{"id":101,"label":987},"21","The counts go 1, 2, 3, 5, 8, 13 for 1 to 6 beats.",{"itemId":990,"prompt":991,"options":992,"correct":750,"why":997},"patterns.deepen-q-meru","In the Meru Prastara, what does row 6 add up to?",[993,994,995,996],{"id":100,"label":644},{"id":747,"label":753},{"id":750,"label":647},{"id":101,"label":650},"Row n adds up to 2ⁿ, so row 6 totals 64 (1 + 6 + 15 + 20 + 15 + 6 + 1).",{"itemId":999,"prompt":1000,"options":1001,"correct":747,"why":1010},"patterns.deepen-q-circle","With 6 points on a circle all joined, the most regions is 31, not 32. What does this show?",[1002,1004,1006,1008],{"id":100,"label":1003},"The count is wrong",{"id":747,"label":1005},"A pattern that fits the first terms can still break",{"id":750,"label":1007},"Doubling never happens in geometry",{"id":101,"label":1009},"Circles have no patterns","The first five answers happen to double, but the true rule is 1 + lines + crossings. Only a proof makes a pattern certain.",{"itemId":1012,"prompt":1013,"options":1014,"correct":100,"why":1020},"patterns.deepen-q-history","Who stated the rhythm-counting rule for the numbers 1, 2, 3, 5, 8, … around 1150, before Fibonacci’s book of 1202?",[1015,1016,1017,1019],{"id":100,"label":501},{"id":747,"label":513},{"id":750,"label":1018},"Gauss",{"id":101,"label":517},"The Jain scholar Hemachandra gave the rule about half a century before Fibonacci; Virahānka and Gopāla had given it earlier still.",{"id":1022,"type":1023,"prompt":1024},"reflect-deepen","reflection","The circle-regions pattern went 1, 2, 4, 8, 16 and then 31. Write a short paragraph for a friend explaining the difference between **noticing a pattern** and **proving** it. Use one pattern you have proved in this layer and one that broke.",{"id":1026,"type":1027,"conceptId":1028,"relation":1029,"explanation":1030},"conn-order","connection","order-of-operations","helps_understand","Formulas like 5n + 4 and n(n + 1) ÷ 2 rely on the order of operations: multiply before adding, and brackets first.",{"id":1032,"type":1027,"conceptId":1033,"relation":1034,"explanation":1035},"conn-properties","properties-of-numbers","related_to","The divisibility test for 9 comes from 10a + b = 9a + (a + b), and expanding brackets uses the distributive property.",{"id":1037,"type":1027,"conceptId":1038,"relation":1034,"explanation":1039},"conn-prime","prime-and-composite","n × n − n + 41 gives primes for n = 1 to 40 and then a composite number: a warning about trusting patterns in primes.",{"id":1041,"type":1042,"title":1043,"points":1044},"cheat-deepen","summary","Cheat sheet",[1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055],"**Algebra:** a variable (n) stands for any position. 3n means 3 × n. **Tₙ** is the nth term. In 5n + 4, 5 is the coefficient and 4 the constant.","**Arithmetic:** Tₙ = a + (n − 1)d = dn + (a − d). The step is the coefficient because the sequence moves in step with the d times table.","**Is x a term?** Solve dn + c = x. If n is a positive whole number, yes. Remainders give a quick test.","**Equivalent expressions** come from different ways of seeing: 4 + 3(n − 1) = 2n + (n + 1) = 3n + 1.","**Geometric:** Tₙ = a × rⁿ⁻¹. 1 + 2 + 4 + … + 2ⁿ⁻¹ = 2ⁿ − 1 (double and subtract).","**Differences:** constant 1st → linear; constant 2nd → quadratic (n²); constant ratio → exponential.","**Proved sums:** 1 + … + n = n(n + 1) ÷ 2; 1 + 3 + … + (2n − 1) = n²; 1³ + … + n³ = (1 + … + n)².","**Fibonacci in India:** rhythms of short and long syllables (Pingala, Virahānka, Gopāla, Hemachandra c. 1150) give 1, 2, 3, 5, 8, … before Fibonacci (1202). Ratios approach φ ≈ 1.618.","**Meru Prastara:** each number is the sum of the two above. Rows total 2ⁿ; diagonals give counting, triangular and tetrahedral numbers; shallow diagonals give Fibonacci.","**Digit patterns** work until a column sum reaches 10 and a carry appears: 111,111,111² is a palindrome; 1,111,111,111² is not.","**Patterns can break:** circle regions 1, 2, 4, 8, 16, 31; n² − n + 41 is prime up to n = 40 only. Examples suggest; proofs decide.",{"id":1057,"type":1058,"sourceIds":1059},"sources-deepen","sources",[1060,1061,1062,1063,1064,1065,1066,1067,1068],"patterns-ncert-ganita-prakash-6","patterns-ncert-class6-algebra","patterns-wiki-fibonacci","patterns-wiki-pascal","patterns-wiki-circle-regions","patterns-mathsisfun-fibonacci","patterns-mathsisfun-sequences","patterns-wiki-pingala","patterns-wiki-aryabhata",[1060,1061,1062,1063,1064,1065,1066,1067,1068],"needs_review",{"generatedBy":1072,"notes":1073},"claude-code","Draft generated locally from a Python script with every number computed and asserted; pending owner review.","9e4d961efd9d983b0e007661be5e63f27d8fe740c1dbbdb484f8aad34e3ce2bc",{"component:pattern-machine@1":1076,"component:match-pairs@1":1077,"logic:practice":1078,"component:sort-game@1":1079,"source:patterns-mathsisfun-fibonacci":1080,"source:patterns-mathsisfun-sequences":1081,"source:patterns-ncert-class6-algebra":1082,"source:patterns-ncert-ganita-prakash-6":1083,"source:patterns-wiki-aryabhata":1084,"source:patterns-wiki-circle-regions":1085,"source:patterns-wiki-fibonacci":1086,"source:patterns-wiki-pascal":1087,"source:patterns-wiki-pingala":1088},"b3d384b49b4d138a00726767a816928da91f44fad70d7d5ccfca6db297c45839","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","b77a2fee9032c27c6a0b903d14287fc8a01076a000aba32f0ffc0e3ff54db124","1c634ad2570a389d1bc96431d0a220aa359ad82b4d33f5b55eb0b12dab8bab4c","b31773e8224c076a8a34aa4e6df704febaac5ecd4d6e1d3ba5041a31aad871e1","36726d3ff16d73627216db999b62b46580dfaf514f313774e79669de57d7c595","aa6cd068c47d9b0a78e3216d27b6aaebc757a552cf3c6e17000a461e400598f8","3036d0a86299354fe4f0f1df748a233c1fe9fdf208b4e75f1169be57492e7095","2b7f65c540dd62068d0f516bd71c5de387803d7540246de06abc95456694188a","8b958ff4286a612ef0e8589f7261e9e18d2878dcfc122f6277212421fcccfa92","1722495dbdbdc18fa17cb0954f2f3a867aa1fd5ad88f44f38f0facfeb8b56cde",{"state":1090,"reviewer":1091,"selfReview":1092,"reviewedAt":1093,"method":1094},"approved","The library owner",true,"2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899598578]