[{"data":1,"prerenderedAt":1047},["ShallowReactive",2],{"layer:patterns:understand":3},{"layer":4,"contentHash":1028,"dependencyHashes":1029,"approval":1040,"releaseId":1046},{"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":1023,"reviewStatus":1024,"authoring":1025},1,"patterns","en","understand","Rules, terms and sequences","Arithmetic and geometric sequences, special numbers, digit patterns and shape rules","Learn the precise language of sequences, the difference method for finding rules, arithmetic and geometric sequences, square, cube, triangular and Fibonacci numbers, digit patterns, and the rules behind growing matchstick and dot patterns.",[13,14,15,16,17],"Use the words sequence, term, position, nth term, term-to-term rule and position-to-term rule correctly.","Find the rule of a sequence using differences and ratios, and tell arithmetic from geometric sequences.","Recognise and extend square, cube, triangular, Fibonacci and alternating sequences.","Explain rules such as 3 × n + 1 for growing matchstick patterns and use them to predict far-away pictures.","Spot and avoid the common mistakes: n − 1 jumps, checking every gap, and mixing up position and term.",45,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Understand",{"label":26,"value":27},"Reading time","≈ 45 minutes",{"label":29,"value":30},"Prior knowledge","Discover: units, differences, jumps",{"label":32,"value":33},"Chapters","11",{"label":35,"value":36},"Labs","Pattern machine ×3, sort game, match pairs",{"label":38,"value":39},"Key formula","first + (n − 1) × difference",[41,45,51,57,60,81,84,165,170,173,206,219,224,229,232,243,252,300,305,308,319,334,339,344,376,453,457,462,465,513,518,522,567,572,575,584,587,600,605,608,644,649,663,668,671,703,715,719,745,769,774,777,789,801,832,837,842,974,978,984,989,993,997,1013],{"id":42,"type":43,"markdown":44},"intro","prose","In Discover you met patterns by looking and building. Now it is time to be **precise**. What exactly is a term? What is the difference between a rule that says “add 4 each time” and one that says “multiply the position by 4 and add 3”? Why do some patterns grow steadily while others explode? Which famous number patterns should every mathematician know by heart?\n\nThis layer gives you the words, the methods and the most common mix-ups. By the end you will be able to look at almost any school-level sequence, name its type, find its rule, and use the rule to predict terms far along without writing them all out.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"how-to","callout","observation","How to use this layer","Each chapter adds one tool. Try the worked examples yourself first, then check. The labs are the place to practise: the pattern machine keeps score, so try to beat your own streak.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch01","chapter","The language of sequences","Chapter 01","1 Words we use",{"id":58,"type":43,"markdown":59},"lang","A **sequence** is a list of numbers (or shapes) in a definite order. Each number in it is a **term**. Each term has a **position**, also called its **term number**: the first term is in position 1, the second in position 2, and so on. Mathematicians use the letter **n** for “the position we are talking about”, so the **nth term** means “the term in position n”.\n\n| Position (n) | 1 | 2 | 3 | 4 | 5 |\n| --- | --- | --- | --- | --- | --- |\n| Term | 7 | 11 | 15 | 19 | 23 |\n\nHere the 1st term is 7 and the 4th term is 19. Do not mix up the position with the term itself: in this sequence the number 11 is in position 2.\n\nA sequence can be **finite** (it stops, like the dates in September: 1 to 30) or **infinite** (it goes on for ever, like the even numbers). When a sequence goes on, we write three dots, **…**, called an *ellipsis*.",{"id":61,"type":62,"tone":63,"items":64},"spec-two-rules","spec","blue",[65,69,73,77],{"label":66,"big":67,"value":68},"Term-to-term rule","add 4","Tells you how to get from one term to the **next**. Easy to use, but slow for far-away terms.",{"label":70,"big":71,"value":72},"Position-to-term rule","4 × n + 3","Tells you how to get a term straight from its **position**. Fast: the 100th term is 4 × 100 + 3 = 403.",{"label":74,"big":75,"value":76},"Start","1st term = 7","A term-to-term rule is useless without a starting term.",{"label":78,"big":79,"value":80},"Check","n = 2 → 11","Always test a position-to-term rule on a term you already know.",{"id":82,"type":43,"markdown":83},"two-rules","There are two ways to describe the rule of a sequence.\n\nA **term-to-term rule** says how each term comes from the one before: *“start at 7, add 4 each time”*. It is how you usually spot a pattern.\n\nA **position-to-term rule** says how each term comes from its position: *“multiply the position by 4, then add 3”*, which we can write as **4 × n + 3**. It is how you predict: to find the 100th term you do not need the 99th, just put n = 100.\n\nThe two rules describe the **same** sequence. Check: position 1 gives 4 × 1 + 3 = 7; position 2 gives 4 × 2 + 3 = 11; position 3 gives 15. ✓ In the Deepen layer you will see exactly why a step of 4 always turns into “4 × n”.",{"id":85,"type":86,"title":87,"terms":88},"glossary-understand","glossary","Sequence vocabulary",[89,93,97,101,105,109,113,117,121,125,129,133,137,141,145,149,153,157,161],{"term":90,"meaning":91,"example":92},"sequence","An ordered list of numbers or shapes, usually made by a rule.","3, 6, 9, 12, …",{"term":94,"meaning":95,"example":96},"term","One number (or picture) in a sequence.","The 3rd term of 3, 6, 9, 12 is 9.",{"term":98,"meaning":99,"example":100},"position (term number)","Where a term sits in the sequence: 1st, 2nd, 3rd, …; often called n.","In 3, 6, 9, 12 the term 12 is in position 4.",{"term":102,"meaning":103,"example":104},"nth term","The term in position n; also a formula that gives any term from its position.","For 3, 6, 9, …, the nth term is 3 × n.",{"term":106,"meaning":107,"example":108},"term-to-term rule","A rule that gets each term from the one before it.","“Add 5 each time.”",{"term":110,"meaning":111,"example":112},"position-to-term rule","A rule that gets a term directly from its position.","“Multiply the position by 5, then subtract 2.”",{"term":114,"meaning":115,"example":116},"difference","A term minus the term before it.","In 2, 9, 16, the difference is 7.",{"term":118,"meaning":119,"example":120},"common difference","The fixed amount added each time in an arithmetic sequence.","5, 8, 11, 14 has common difference 3.",{"term":122,"meaning":123,"example":124},"arithmetic sequence","A sequence where the same number is added (or subtracted) each time.","20, 17, 14, 11, …",{"term":126,"meaning":127,"example":128},"ratio","A term divided by the term before it.","In 3, 12, 48, the ratio is 4.",{"term":130,"meaning":131,"example":132},"common ratio","The fixed number you multiply by each time in a geometric sequence.","2, 6, 18, 54 has common ratio 3.",{"term":134,"meaning":135,"example":136},"geometric sequence","A sequence where each term is the one before multiplied by the same number.","1, 10, 100, 1,000, …",{"term":138,"meaning":139,"example":140},"square number","A number that is a whole number times itself; the dots in a square array.","49 = 7 × 7",{"term":142,"meaning":143,"example":144},"cube number","A number that is a whole number times itself three times; the small cubes in a big cube.","27 = 3 × 3 × 3",{"term":146,"meaning":147,"example":148},"triangular number","The sum 1 + 2 + 3 + … + n; the dots in a triangle.","1 + 2 + 3 + 4 = 10",{"term":150,"meaning":151,"example":152},"Fibonacci sequence","The sequence 1, 1, 2, 3, 5, 8, … where each term is the sum of the two before it.","13 + 21 = 34",{"term":154,"meaning":155,"example":156},"alternating pattern","A pattern whose step changes back and forth between two (or more) values.","+3, −1, +3, −1: 1, 4, 3, 6, 5, …",{"term":158,"meaning":159,"example":160},"finite \u002F infinite","A finite sequence stops; an infinite sequence goes on for ever.","Days of a month are finite; odd numbers are infinite.",{"term":162,"meaning":163,"example":164},"ellipsis","The three dots … that mean “and so on”.","1, 2, 3, …",{"id":166,"type":53,"title":167,"eyebrow":168,"navLabel":169},"ch02","Finding the rule: the difference method","Chapter 02","2 Finding the rule",{"id":171,"type":43,"markdown":172},"diff-method","The first thing to try with any number sequence is the **difference method**: subtract each term from the next, and write the differences in a row underneath.\n\nWhat the differences tell you:\n\n1. **All the same** → the sequence adds a fixed amount: an **arithmetic sequence**. The rule is “add the difference”.\n2. **They go up by the same amount** (like 3, 5, 7, 9) → a **square-number-like** sequence. Look for n × n hidden inside it.\n3. **They grow in proportion to the terms** (the differences are a copy of the sequence, or a multiple of it) → try **dividing** instead: it may be a **geometric sequence**.\n4. **They take turns** (like +3, −1, +3, −1) → an **alternating** pattern.\n5. **Each term is the sum of the two before** → a **Fibonacci-type** sequence.\n\nIf none of these fit, try the **ratio** (divide each term by the one before), look at the digits, or think about pictures.",{"id":174,"type":175,"caption":176,"columns":177,"rows":181},"table-diffs","table","Differences reveal the type (computed)",[178,179,180],"Sequence","Differences","What it tells you",[182,186,190,194,198,202],[183,184,185],"5, 8, 11, 14, 17, 20","3, 3, 3, 3, 3","Same each time: arithmetic, add 3",[187,188,189],"1, 4, 9, 16, 25, 36","3, 5, 7, 9, 11","Differences go up by 2: square numbers",[191,192,193],"1, 3, 6, 10, 15, 21","2, 3, 4, 5, 6","Differences go up by 1: triangular numbers",[195,196,197],"3, 6, 12, 24, 48, 96","3, 6, 12, 24, 48","Differences are the sequence again: geometric, × 2",[199,200,201],"1, 1, 2, 3, 5, 8, 13","0, 1, 1, 2, 3, 5","Differences are the Fibonacci numbers again",[203,204,205],"1, 4, 3, 6, 5, 8","+3, −1, +3, −1, +3","Take turns: alternating, +3 then −1",{"id":207,"type":208,"title":209,"problem":210,"steps":211,"help":217},"we-find-rule","worked_example","Find the rule and the next two terms","Find the rule and the next two terms of **2, 6, 12, 20, 30, …**",[212,213,214,215,216],"Differences: 6 − 2 = 4, 12 − 6 = 6, 20 − 12 = 8, 30 − 20 = 10.","The differences are 4, 6, 8, 10: they go up by 2. So the next differences are **12** and **14**.","Next terms: 30 + 12 = **42**, then 42 + 14 = **56**.","Spot the hidden rule: 2 = 1 × 2, 6 = 2 × 3, 12 = 3 × 4, 20 = 4 × 5, 30 = 5 × 6. Each term is **n × (n + 1)**.","Check: 6 × 7 = 42 ✓ and 7 × 8 = 56 ✓. These are the **rectangular** (or *pronic*) numbers, exactly twice the triangular numbers.",{"simplerExplanation":218},"Write how much it goes up by each time: 4, 6, 8, 10. That is going up by 2, so next it goes up by 12, then 14.",{"id":220,"type":47,"variant":221,"title":222,"markdown":223},"mis-two-terms","misconception","“I checked the first two terms, so my rule is right”","For **3, 6, 12, …**, the rule “add 3” fits the first two terms (3 → 6) but not the third (6 + 3 = 9, not 12). The rule “double” fits all three. A rule must work for **every** term you have. Check each gap, not just the first.",{"id":225,"type":53,"title":226,"eyebrow":227,"navLabel":228},"ch03","Arithmetic sequences: add the same amount","Chapter 03","3 Add a constant",{"id":230,"type":43,"markdown":231},"arith","An **arithmetic sequence** (say *a-rith-MET-ic*, with the stress on MET) goes up or down by the **same amount** every time. That fixed amount is the **common difference**.\n\n- 4, 9, 14, 19, 24, … common difference **+5**\n- 100, 93, 86, 79, … common difference **−7** (a *decreasing* arithmetic sequence)\n- 0.5, 1, 1.5, 2, … common difference **+0.5**\n- −6, −2, 2, 6, 10, … common difference **+4**, starting below zero\n\nArithmetic sequences appear whenever something changes at a **steady rate**: saving the same amount every week, a taxi meter adding the same fare every kilometre, a candle burning down 2 cm every hour, or the seat numbers along a row.\n\nTo find a far-away term, count the **jumps**. From the 1st term to the nth term there are **n − 1** jumps. So:\n\n**nth term = first term + (n − 1) × common difference**",{"id":233,"type":208,"title":234,"problem":235,"steps":236,"help":241},"we-auto","The auto-rickshaw meter","An auto-rickshaw fare is **₹30 for the first kilometre** and **₹15 for each extra kilometre**. Write the fares for 1, 2, 3, 4 km and find the fare for a **12 km** ride.",[237,238,239,240],"Fares: 1 km ₹30, 2 km ₹45, 3 km ₹60, 4 km ₹75. The common difference is ₹15.","From 1 km to 12 km there are 12 − 1 = **11** extra kilometres (jumps).","Fare = 30 + 11 × 15 = 30 + 165 = **₹195**.","Check with the position rule: fare = 15 × n + 15. For n = 12: 180 + 15 = 195. ✓",{"simplerExplanation":242},"The first kilometre costs ₹30. The other 11 kilometres cost ₹15 each, which is ₹165. Together: ₹195.",{"id":244,"type":208,"title":245,"problem":246,"steps":247},"we-decreasing","A sequence going down","Find the **10th term** of 100, 93, 86, 79, … and decide whether **2** is a term of this sequence.",[248,249,250,251],"Common difference: 93 − 100 = −7. We subtract 7 each time.","10th term: 100 + 9 × (−7) = 100 − 63 = **37**.","Is 2 a term? The terms are 100 minus a multiple of 7. 100 − 2 = 98 and 98 = 7 × 14, so yes: 2 is the term after 14 jumps, the **15th term**.","Check: 100 − 14 × 7 = 100 − 98 = 2. ✓ The next term would be 2 − 7 = −5, below zero.",{"id":253,"type":254,"component":255,"componentVersion":5,"config":256,"objective":293,"textAlternative":294,"help":295},"lab-arith-geo","interactive","pattern-machine",{"puzzles":257},[258,265,270,274,278,283,285,289],{"kind":259,"rule":260,"show":262,"ask":264},"number",{"type":261,"start":262,"step":263},"add",4,5,2,{"kind":259,"rule":266,"show":262,"ask":264,"hint":269},{"type":261,"start":267,"step":268},100,-7,"Subtract to find the step. It is negative.",{"kind":259,"rule":271,"show":262,"ask":264,"hint":273},{"type":261,"start":272,"step":272},0.5,"Steps can be halves.",{"kind":259,"rule":275,"show":262,"ask":264,"hint":277},{"type":261,"start":276,"step":262},-6,"Numbers below zero follow the same rules.",{"kind":259,"rule":279,"show":262,"ask":264,"hint":282},{"type":280,"start":281,"factor":264},"multiply",3,"Divide a term by the one before it.",{"kind":259,"rule":284,"show":263,"ask":264},{"type":280,"start":5,"factor":281},{"kind":259,"rule":286,"show":262,"ask":5,"hint":288},{"type":280,"start":263,"factor":287},10,"Place value: each digit moves one place left.",{"kind":259,"rule":290,"show":262,"ask":264,"hint":292},{"type":280,"start":291,"factor":272},256,"Multiplying by a half is the same as halving.","Decide whether each sequence adds or multiplies, then predict the next terms and choose the rule.","The pattern machine shows the first terms of eight sequences, asks for the next ones, and then asks you to choose the rule from four options.\n\nArithmetic (add the same amount):\n\n1. 4, 9, 14, 19, … next **24, 29** (add 5).\n2. 100, 93, 86, 79, … next **72, 65** (subtract 7).\n3. 0.5, 1, 1.5, 2, … next **2.5, 3** (add 0.5).\n4. −6, −2, 2, 6, … next **10, 14** (add 4).\n\nGeometric (multiply by the same number):\n\n5. 3, 6, 12, 24, … next **48, 96** (multiply by 2).\n6. 1, 3, 9, 27, 81, … next **243, 729** (multiply by 3).\n7. 5, 50, 500, 5,000, … next **50,000** (multiply by 10).\n8. 256, 128, 64, 32, … next **16, 8** (multiply by ½, which is halving).\n\nTo decide which kind: subtract neighbours. Equal differences mean arithmetic. If not, divide neighbours. Equal ratios mean geometric.",{"simplerExplanation":296,"hints":297},"First try taking away neighbours. If the answers are all the same, keep adding that. If not, try dividing.",[298,299],"For 3, 6, 12, 24 the differences are 3, 6, 12: not equal. But 6 ÷ 3 = 12 ÷ 6 = 2.","A negative step means the numbers go down.",{"id":301,"type":53,"title":302,"eyebrow":303,"navLabel":304},"ch04","Geometric sequences: multiply by the same number","Chapter 04","4 Multiply",{"id":306,"type":43,"markdown":307},"geo","In a **geometric sequence** each term is the one before **multiplied by the same number**, the **common ratio**.\n\n- 1, 2, 4, 8, 16, 32, … ratio **2** (the *powers of 2*)\n- 1, 3, 9, 27, 81, … ratio **3** (the *powers of 3*)\n- 1, 10, 100, 1,000, 10,000, … ratio **10**: this is our place-value system! Ones, tens, hundreds, thousands.\n- 64, 32, 16, 8, 4, … ratio **½** (halving)\n- 81, 27, 9, 3, 1, … ratio **⅓**\n\nGeometric sequences describe things that **multiply**: a message forwarded to 3 new people by everyone who receives it; bacteria that split in two every 20 minutes; a sheet of paper folded in half again and again (2, 4, 8, 16 layers); a ball that bounces to half its height each time.\n\nTo find the ratio, **divide** a term by the one before it: 27 ÷ 9 = 3. The nth term is **first term × ratio × ratio × … (n − 1 times)**.",{"id":309,"type":208,"title":310,"problem":311,"steps":312,"help":317},"we-forward","How fast does a message spread?","Riya sends a message to **3** friends (round 1). Each of them forwards it to 3 new friends (round 2), and so on. How many new people receive it in **round 6**, and how many people have received it altogether after 6 rounds?",[313,314,315,316],"Round 1: 3 people. Round 2: 3 × 3 = 9. Round 3: 27. Round 4: 81. Round 5: 243. Round 6: **729**.","This is geometric with first term 3 and ratio 3: 3, 9, 27, 81, 243, 729.","Altogether: 3 + 9 + 27 + 81 + 243 + 729 = **1092** people after only six rounds.","Compare with an arithmetic rule “3 new people each round”: only 3 × 6 = 18 people. Multiplying grows **much** faster than adding.",{"anotherExample":318},"Paper folding: 1 sheet folded once is 2 layers, then 4, 8, 16, 32. After 10 folds there would be 1,024 layers (if you could fold it that many times!).",{"id":320,"type":321,"prompt":322,"options":323,"explanation":333},"predict-race","prediction","Two plans for pocket money over 10 days. **Plan A:** ₹100 on day 1, then ₹100 more each day (₹100, ₹200, ₹300, …). **Plan B:** ₹1 on day 1, then double each day (₹1, ₹2, ₹4, …). On **day 10**, which plan pays more *that day*?",[324,327,330],{"id":325,"label":326},"a","Plan A pays more on day 10",{"id":328,"label":329},"b","Plan B pays more on day 10",{"id":331,"label":332},"c","They pay the same on day 10","**Plan A, but only just, and not for long.** On day 10, Plan A pays 100 × 10 = **₹1000**, while Plan B pays 1 doubled 9 times = **₹512**. On day 11 Plan B pays ₹1,024, still a little less than Plan A's ₹1,100. On day 12 Plan B pays ₹2,048 against ₹1,200 and has overtaken for good. By day 20 Plan B pays ₹524,288 in a single day while Plan A pays ₹2,000. Multiplying always beats adding in the end.",{"id":335,"type":47,"variant":336,"title":337,"markdown":338},"nuance-geo-down","nuance","Geometric sequences can shrink","If the ratio is between 0 and 1 (like ½ or ⅓), the terms get smaller and smaller but **never reach zero**: 64, 32, 16, 8, 4, 2, 1, ½, ¼, ⅛, … A bouncing ball that rises to half its height each time does this, at least in an ideal world. A decreasing arithmetic sequence is different: it falls by the same amount each time and soon crosses zero into negative numbers.",{"id":340,"type":53,"title":341,"eyebrow":342,"navLabel":343},"ch05","Arithmetic or geometric? Telling them apart","Chapter 05","5 Add or multiply?",{"id":345,"type":175,"caption":346,"columns":347,"rows":351},"table-compare","Arithmetic and geometric sequences compared",[348,349,350],"Feature","Arithmetic","Geometric",[352,356,360,364,368,372],[353,354,355],"Each step","Add the same number","Multiply by the same number",[357,358,359],"Test","Differences are equal","Ratios are equal",[361,362,363],"Example","3, 7, 11, 15, 19 (add 4)","3, 6, 12, 24, 48 (× 2)",[365,366,367],"Picture","A straight staircase: equal steps","A curve that gets steeper and steeper",[369,370,371],"10th term of the example","3 + 9 × 4 = 39","3 × 2⁹ = 1,536",[373,374,375],"Real life","Taxi fares, weekly savings, seat numbers","Doubling bacteria, place value, forwarded messages",{"id":377,"type":254,"component":378,"componentVersion":5,"config":379,"objective":447,"textAlternative":448,"help":449},"lab-sort-ag","sort-game",{"prompt":380,"bins":381,"items":387,"seconds":446},"Sort each sequence: arithmetic (add a constant), geometric (multiply by a constant), or neither.",[382,383,384],{"id":230,"label":349},{"id":306,"label":350},{"id":385,"label":386},"neither","Neither",[388,392,396,400,404,408,412,416,420,424,428,431,435,438,442],{"id":389,"label":390,"bin":230,"why":391},"s1","5, 11, 17, 23, 29","Differences are all 6.",{"id":393,"label":394,"bin":306,"why":395},"s2","2, 10, 50, 250, 1,250","Each term is 5 times the one before.",{"id":397,"label":398,"bin":385,"why":399},"s3","1, 4, 9, 16, 25","Square numbers: differences 3, 5, 7, 9 are not equal and ratios 4, 2.25, … are not equal.",{"id":401,"label":402,"bin":230,"why":403},"s4","90, 81, 72, 63, 54","Subtract 9 each time: common difference −9.",{"id":405,"label":406,"bin":306,"why":407},"s5","1, 10, 100, 1,000, 10,000","Multiply by 10 each time: the place values.",{"id":409,"label":410,"bin":385,"why":411},"s6","1, 1, 2, 3, 5, 8","Fibonacci: each term is the sum of the two before, not a fixed step or ratio.",{"id":413,"label":414,"bin":230,"why":415},"s7","₹30, ₹45, ₹60, ₹75 (auto fares)","Add ₹15 per kilometre.",{"id":417,"label":418,"bin":306,"why":419},"s8","400, 200, 100, 50, 25","Halve each time: ratio ½.",{"id":421,"label":422,"bin":385,"why":423},"s9","1, 3, 6, 10, 15","Triangular numbers: the differences 2, 3, 4, 5 keep growing by 1.",{"id":425,"label":426,"bin":230,"why":427},"s10","0.2, 0.4, 0.6, 0.8, 1.0","Add 0.2 each time.",{"id":429,"label":184,"bin":230,"why":430},"s11","Add 0 each time. (It is also geometric with ratio 1: a sequence that fits both!)",{"id":432,"label":433,"bin":306,"why":434},"s12","2, 6, 18, 54, 162","Multiply by 3 each time.",{"id":436,"label":203,"bin":385,"why":437},"s13","Alternating: +3, −1, +3, −1. The step is not constant.",{"id":439,"label":440,"bin":230,"why":441},"s14","−8, −5, −2, 1, 4","Add 3 each time, starting below zero.",{"id":443,"label":444,"bin":385,"why":445},"s15","1, 8, 27, 64, 125","Cube numbers: neither the differences nor the ratios are equal.",0,"Tell arithmetic, geometric and other sequences apart using differences and ratios.","A sorting game with 15 sequence cards and three bins.\n\n**Arithmetic:** 5, 11, 17, 23, 29 (add 6); 90, 81, 72, 63, 54 (subtract 9); auto fares ₹30, ₹45, ₹60, ₹75 (add 15); 0.2, 0.4, 0.6, 0.8, 1.0 (add 0.2); −8, −5, −2, 1, 4 (add 3); and 3, 3, 3, 3, 3 (add 0, which is also geometric with ratio 1).\n\n**Geometric:** 2, 10, 50, 250, 1,250 (× 5); 1, 10, 100, 1,000, 10,000 (× 10); 400, 200, 100, 50, 25 (× ½); 2, 6, 18, 54, 162 (× 3).\n\n**Neither:** square numbers 1, 4, 9, 16, 25; Fibonacci 1, 1, 2, 3, 5, 8; triangular numbers 1, 3, 6, 10, 15; the alternating pattern 1, 4, 3, 6, 5, 8; and cube numbers 1, 8, 27, 64, 125.\n\nMethod: check the differences first, then the ratios. If neither is constant, the sequence belongs to another family.",{"hints":450},[451,452],"Subtract neighbours. All equal? Arithmetic.","Divide neighbours. All equal? Geometric.",{"id":454,"type":47,"variant":221,"title":455,"markdown":456},"mis-double-vs-add","“Doubling means adding 2”","**Doubling** means multiplying by 2: 5 → 10 → 20 → 40. **Adding 2** gives 5 → 7 → 9 → 11. They are completely different rules. After 10 steps from 5, doubling reaches 5 × 2¹⁰ = 5,120, while adding 2 reaches only 25.",{"id":458,"type":53,"title":459,"eyebrow":460,"navLabel":461},"ch06","Special number sequences: squares, cubes and triangles","Chapter 06","6 Special numbers",{"id":463,"type":43,"markdown":464},"special","Some sequences are so useful that they have their own names. You should know the first ten or so of each by heart.\n\n**Square numbers** are made by multiplying a number by itself: 1 × 1, 2 × 2, 3 × 3, … We write 5 × 5 as **5²** (“five squared”). They are the numbers of dots in square arrays.\n\n**Cube numbers** are made by multiplying a number by itself three times: 2 × 2 × 2 = **2³** = 8 (“two cubed”). They are the numbers of small cubes needed to build a bigger cube: a 3 by 3 by 3 Rubik's cube is made of 27 little cubes.\n\n**Triangular numbers** are 1, 1 + 2, 1 + 2 + 3, 1 + 2 + 3 + 4, … They are the numbers of dots in triangles, the pins in bowling, and the number of handshakes in a group (if 5 people all shake hands, there are 1 + 2 + 3 + 4 = 10 handshakes).",{"id":466,"type":175,"caption":467,"columns":468,"rows":473},"table-special","The first ten of each special sequence (computed)",[469,470,471,472],"n","Square n × n","Cube n × n × n","Triangular 1 + … + n",[474,476,481,485,489,494,498,503,505,509],[475,475,475,475],"1",[477,478,479,480],"2","4","8","3",[480,482,483,484],"9","27","6",[478,486,487,488],"16","64","10",[490,491,492,493],"5","25","125","15",[484,495,496,497],"36","216","21",[499,500,501,502],"7","49","343","28",[479,487,504,495],"512",[482,506,507,508],"81","729","45",[488,510,511,512],"100","1,000","55",{"id":514,"type":47,"variant":515,"title":516,"markdown":517},"aha-squares-odds","aha","Squares are built from odd numbers","Look at the differences between square numbers: 4 − 1 = 3, 9 − 4 = 5, 16 − 9 = 7, 25 − 16 = 9. They are the **odd numbers**! So:\n\n- 1 = 1\n- 1 + 3 = 4\n- 1 + 3 + 5 = 9\n- 1 + 3 + 5 + 7 = 16\n\nAdding the first n odd numbers always gives **n × n**. In the Investigate layer you will see a picture that shows why: each odd number is an L-shaped layer wrapped around a square.",{"id":519,"type":47,"variant":515,"title":520,"markdown":521},"aha-two-triangles","Two neighbouring triangular numbers make a square","3 + 6 = 9, 6 + 10 = 16, 10 + 15 = 25. Put two dot triangles of neighbouring sizes together and they fit into a perfect square. Try it with counters: a triangle of 6 and a triangle of 10 make a 4 by 4 square.",{"id":523,"type":254,"component":255,"componentVersion":5,"config":524,"objective":561,"textAlternative":562,"help":563},"lab-special",{"puzzles":525},[526,529,533,537,542,545,552,556,559],{"kind":259,"rule":527,"show":263,"ask":264},{"type":528,"offset":446},"squares",{"kind":259,"rule":530,"show":263,"ask":264,"hint":532},{"type":531},"triangular","Differences: 2, 3, 4, 5, …",{"kind":259,"rule":534,"show":262,"ask":5,"hint":536},{"type":535},"cubes","1 × 1 × 1, 2 × 2 × 2, …",{"kind":259,"rule":538,"show":540,"ask":264,"hint":541},{"type":539,"a":5,"b":5},"fibonacci",7,"Add the two terms before.",{"kind":259,"rule":543,"show":263,"ask":264,"hint":544},{"type":528,"offset":5},"Compare each term with a square number.",{"kind":259,"rule":546,"show":550,"ask":264,"hint":551},{"type":547,"start":5,"steps":548},"alternate",[281,549],-1,6,"The step changes back and forth.",{"kind":259,"rule":553,"show":550,"ask":264},{"type":547,"start":287,"steps":554},[263,555],-2,{"kind":259,"rule":557,"show":550,"ask":264,"hint":558},{"type":539,"a":264,"b":5},"Same rule as Fibonacci, different start.",{"kind":259,"rule":560,"show":263,"ask":5},{"type":528,"offset":549},"Recognise square, cube, triangular, Fibonacci and alternating sequences and predict their next terms.","Nine puzzles about the special sequences. After predicting, you choose the rule from four options.\n\n1. 1, 4, 9, 16, 25, … next **36, 49** (square numbers).\n2. 1, 3, 6, 10, 15, … next **21, 28** (triangular numbers; add 6, then 7).\n3. 1, 8, 27, 64, … next **125** (cube numbers).\n4. 1, 1, 2, 3, 5, 8, 13, … next **21, 34** (Fibonacci: each term is the sum of the two before).\n5. 2, 5, 10, 17, 26, … next **37, 50** (square numbers plus 1).\n6. 1, 4, 3, 6, 5, 8, … next **7, 10** (add 3, subtract 1, repeat).\n7. 10, 15, 13, 18, 16, 21, … next **19, 24** (add 5, subtract 2, repeat).\n8. 2, 1, 3, 4, 7, 11, … next **18, 29** (the Lucas numbers: Fibonacci's rule with a different start).\n9. 0, 3, 8, 15, 24, … next **35** (square numbers minus 1).",{"hints":564},[565,566],"Compare each term with the square numbers 1, 4, 9, 16, 25.","If the differences go up and down, the rule is alternating.",{"id":568,"type":53,"title":569,"eyebrow":570,"navLabel":571},"ch07","Fibonacci and alternating patterns","Chapter 07","7 Fibonacci & friends",{"id":573,"type":43,"markdown":574},"fib","Here is a sequence with a different kind of rule: **1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …**\n\nEach term is the **sum of the two terms before it**: 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, and so on. It is called the **Fibonacci sequence**, after the Italian mathematician Leonardo of Pisa (nicknamed Fibonacci), who wrote about it in 1202 in a puzzle about breeding rabbits. But Indian scholars knew it centuries earlier: poets and musicians counting the rhythms that can be made from short and long beats found exactly these numbers. The NCERT textbook calls them **Virahānka numbers**, after one of those scholars. You will meet the full story, including Pingala and Hemachandra, in the Deepen layer.\n\nThe Fibonacci numbers turn up in nature surprisingly often: the spirals of seeds in a sunflower head (often 34 and 55, or 55 and 89), the spirals on a pineapple or a pinecone (often 8 and 13), and the number of petals on many flowers (3, 5, 8 or 13 are common).\n\nSay that carefully, though. It is a strong tendency, not a rule of nature. When scientists asked the public to photograph and count 657 sunflower heads, about four counts in five came out as Fibonacci numbers — and nearly one in five did not. So if you count a real pinecone and get 7 and 11, your counting is probably fine; the plant simply did something else.",{"id":576,"type":208,"title":577,"problem":578,"steps":579},"we-fib","Continue a Fibonacci-type sequence","A sequence follows the Fibonacci rule (each term is the sum of the two before) and starts **3, 4, …**. Find the first eight terms.",[580,581,582,583],"3rd term: 3 + 4 = 7.","4th: 4 + 7 = 11. 5th: 7 + 11 = 18. 6th: 11 + 18 = 29.","7th: 18 + 29 = 47. 8th: 29 + 47 = **76**.","Sequence: 3, 4, 7, 11, 18, 29, 47, 76. Any two starting numbers give a Fibonacci-type sequence.",{"id":585,"type":43,"markdown":586},"alternating","An **alternating pattern** changes its step back and forth. Look at 1, 4, 3, 6, 5, 8, 7, …: the steps are **+3, −1, +3, −1, …**. Every two steps, the pattern moves up by 3 − 1 = 2 overall.\n\nAlternating patterns can hide two simpler sequences inside them. In 1, 4, 3, 6, 5, 8, 7, 10, … the terms in odd positions are 1, 3, 5, 7 (odd numbers) and the terms in even positions are 4, 6, 8, 10 (even numbers from 4). Splitting a tricky sequence into **every other term** is a powerful trick.\n\nAnother kind of alternating pattern flips the **sign**: 1, −1, 1, −1, … or 1, −2, 3, −4, 5, …",{"id":588,"type":589,"itemId":590,"prompt":591,"check":592,"hints":594,"feedback":597},"pr-alt","practice","patterns.understand-alternate","The pattern goes **2, 7, 5, 10, 8, 13, 11, …** (add 5, subtract 2, and repeat). What is the **10th** term?",{"kind":259,"answer":593,"tolerance":446},19,[595,596],"Continue the pattern: after 11 comes 11 + 5.","Or: every two steps the pattern goes up by 3.",{"correct":598,"incorrect":599},"Yes: 2, 7, 5, 10, 8, 13, 11, 16, 14, 19. The 10th term is 19.","Keep going carefully: 11 + 5 = 16, 16 − 2 = 14, 14 + 5 = 19. The 10th term is 19.",{"id":601,"type":53,"title":602,"eyebrow":603,"navLabel":604},"ch08","Patterns in tables and digits","Chapter 08","8 Digit patterns",{"id":606,"type":43,"markdown":607},"mult-table","The **multiplication table** (the 10 by 10 grid of times tables) is full of patterns:\n\n- It is **symmetric** about the diagonal from top-left to bottom-right: 3 × 7 and 7 × 3 are in mirror-image places and are equal.\n- That diagonal holds the **square numbers**: 1, 4, 9, 16, …, 100.\n- The **5s** row ends in 5, 0, 5, 0, … and the **10s** row ends in 0.\n- The **2s, 4s, 6s, 8s** and **10s** rows contain only even numbers.\n- Every row is an **arithmetic sequence**: the 7s row adds 7.\n- In any row, the last digits repeat: the 4s row ends in 4, 8, 2, 6, 0, 4, 8, 2, 6, 0.",{"id":609,"type":175,"caption":610,"columns":611,"rows":615},"table-digits","Digit patterns (every line computed)",[612,613,614],"Pattern","Lines","What to notice",[616,620,624,628,632,636,640],[617,618,619],"Repunits squared","1 × 1 = 1; 11 × 11 = 121; 111 × 111 = 12,321; 1,111 × 1,111 = 1,234,321","Counts up to the number of 1s, then back down",[621,622,623],"… up to nine 1s","111,111,111 × 111,111,111 = 12,345,678,987,654,321","The “palindrome” (reads the same both ways) peaks at 9",[625,626,627],"× 9 + next","1 × 9 + 2 = 11; 12 × 9 + 3 = 111; 123 × 9 + 4 = 1,111","The answer is all 1s, one more each line",[629,630,631],"× 8 + next","1 × 8 + 1 = 9; 12 × 8 + 2 = 98; 123 × 8 + 3 = 987","Digits count down from 9",[633,634,635],"9s and 8s","9 × 9 + 7 = 88; 98 × 9 + 6 = 888; 987 × 9 + 5 = 8,888","The answer is all 8s",[637,638,639],"Magic 37","37 × 3 = 111; 37 × 6 = 222; 37 × 9 = 333; … 37 × 27 = 999","Multiples of 3 give triple digits",[641,642,643],"9 times table","9, 18, 27, 36, 45, 54, 63, 72, 81, 90","Digits add to 9; tens go up, units go down",{"id":645,"type":47,"variant":646,"title":647,"markdown":648},"careful-indian-commas","careful","Where do the commas go?","In the Indian system 1234321 is written **12,34,321** (twelve lakh thirty-four thousand three hundred twenty-one); in the international system it is **1,234,321** (one million …). The digits are identical, only the grouping differs. In the big line of the table, 12345678987654321 is written with international commas; check the Number system topic if you want to write it the Indian way.",{"id":650,"type":321,"prompt":651,"options":652,"explanation":662},"predict-repunit","You know that 11 × 11 = 121, 111 × 111 = 12,321 and so on up to 111,111,111 × 111,111,111 = 12,345,678,987,654,321. What do you think **1,111,111,111 × 1,111,111,111** (ten 1s) gives?",[653,655,657,659],{"id":325,"label":654},"1234567890987654321",{"id":328,"label":656},"12345678910987654321",{"id":331,"label":658},"Something that breaks the pattern",{"id":660,"label":661},"d","1111111111","**It breaks the pattern.** The real answer is **1234567900987654321**. There is no single digit “10”, so when the columns add up to 10 there is a carry, and the neat counting-up-and-down shape collapses. The pattern worked for nine lines because every column sum was 9 or less. This is your first taste of a pattern that looks certain and then fails. You will see *why* in the Deepen layer.",{"id":664,"type":53,"title":665,"eyebrow":666,"navLabel":667},"ch09","Shape patterns and their rules","Chapter 09","9 Shape rules",{"id":669,"type":43,"markdown":670},"shape-rules","Growing shape patterns turn into number sequences as soon as you count something: sticks, dots, tiles. Then you can find a rule, and the rule lets you predict any picture.\n\nTake **matchstick squares in a row**: 4, 7, 10, 13, … sticks. Two ways to see the rule:\n\n- **Term-to-term:** each new square needs 3 more sticks (top, bottom, and one side). Add 3.\n- **Position-to-term:** start with 1 stick standing on its own at the left, then each square adds 3 sticks (a top, a bottom and a right side). With n squares that is **1 + 3 × n**, which we usually write **3 × n + 1**.\n\nCheck: n = 1 gives 4 ✓; n = 4 gives 13 ✓. So 50 squares need 3 × 50 + 1 = **151** sticks.",{"id":672,"type":175,"caption":673,"columns":674,"rows":678},"table-shape-rules","Growing shape patterns and their rules (checked against counts for n = 1 to 5)",[612,675,676,677],"Counts for n = 1, 2, 3, 4, 5","Rule","n = 10",[679,684,687,692,697,700],[680,681,682,683],"Matchstick squares in a row","4, 7, 10, 13, 16","3 × n + 1","31",[685,188,686,497],"Matchstick triangles in a row","2 × n + 1",[688,689,690,691],"Matchstick hexagons in a row","6, 11, 16, 21, 26","5 × n + 1","51",[693,694,695,696],"Growing L-shapes (tiles)","1, 3, 5, 7, 9","2 × n − 1","19",[698,398,699,510],"Square dot patterns","n × n",[701,422,702,512],"Triangle dot patterns \u002F staircases","n × (n + 1) ÷ 2",{"id":704,"type":208,"title":705,"problem":706,"steps":707,"help":713},"we-hexagons","Why hexagons need 5 × n + 1 sticks","Hexagons made from matchsticks are joined in a row, each sharing one side with the one before. Explain the rule **5 × n + 1** and find the number of sticks for **20 hexagons**.",[708,709,710,711,712],"One hexagon on its own has 6 sides: 6 sticks.","Each new hexagon shares one side with the last, so it needs only 6 − 1 = **5** new sticks.","Think of 1 stick at the left end, then 5 new sticks for each hexagon: 1 + 5 × n.","Check: n = 1 → 6 ✓, n = 2 → 11 ✓, n = 3 → 16 ✓.","For 20 hexagons: 5 × 20 + 1 = **101** sticks.",{"simplerExplanation":714},"Every hexagon after the first needs 5 more sticks. So the counts go 6, 11, 16, 21, … adding 5 each time.",{"id":716,"type":47,"variant":515,"title":717,"markdown":718},"aha-step-multiplier","The step becomes the multiplier","Look at the table: the pattern that **adds 3** has rule **3 × n + 1**; the one that **adds 2** has **2 × n + 1**; the one that **adds 5** has **5 × n + 1**. Whenever a pattern adds the same amount each time, that amount is the number that multiplies n. The number added or subtracted at the end is a correction so that the first term comes out right.",{"id":720,"type":254,"component":255,"componentVersion":5,"config":721,"objective":739,"textAlternative":740,"help":741},"lab-shapes",{"puzzles":722},[723,726,728,731,734,737],{"kind":724,"shape":725,"show":281,"askTerm":287},"shape","matchstick-squares",{"kind":724,"shape":727,"show":281,"askTerm":287},"matchstick-triangles",{"kind":724,"shape":729,"show":281,"askTerm":287,"hint":730},"hexagon-chain","Each new hexagon shares one side.",{"kind":724,"shape":732,"show":262,"askTerm":733},"l-shapes",12,{"kind":724,"shape":735,"show":262,"askTerm":287,"hint":736},"dot-triangles","Add 1 + 2 + 3 + … + 10.",{"kind":724,"shape":738,"show":262,"askTerm":733},"dot-squares","Predict the 10th (or 12th) picture of a growing shape pattern and choose its rule in terms of n.","The machine draws the first pictures of six growing shape patterns, asks for the count in a far picture, then asks you to choose the rule.\n\n1. Matchstick squares: 4, 7, 10. The 10th picture needs **31** sticks. Rule: 3 × n + 1.\n2. Matchstick triangles: 3, 5, 7. The 10th needs **21** sticks. Rule: 2 × n + 1.\n3. Matchstick hexagons: 6, 11, 16. The 10th needs **51** sticks. Rule: 5 × n + 1.\n4. L-shapes: 1, 3, 5, 7 tiles. The 12th needs **23** tiles. Rule: 2 × n − 1.\n5. Dot triangles: 1, 3, 6, 10. The 10th has **55** dots. Rule: n × (n + 1) ÷ 2.\n6. Dot squares: 1, 4, 9, 16. The 12th has **144** dots. Rule: n × n.\n\nThe wrong answers the machine offers are the common slips: for example 4 × n for matchstick squares (forgetting shared sides) or 3 × n + 4 (using the first term instead of the correction).",{"hints":742},[743,744],"The number added each time multiplies n.","Test your rule with n = 1: does it give the first picture?",{"id":746,"type":254,"component":747,"componentVersion":5,"config":748,"objective":767,"textAlternative":768},"lab-match-shape","match-pairs",{"prompt":749,"mode":750,"pairs":751},"Match each growing shape pattern to its rule.","connect",[752,754,756,758,760,762,764],{"a":753,"b":682},"Matchstick squares in a row: 4, 7, 10, …",{"a":755,"b":686},"Matchstick triangles in a row: 3, 5, 7, …",{"a":757,"b":690},"Matchstick hexagons in a row: 6, 11, 16, …",{"a":759,"b":695},"L-shapes of tiles: 1, 3, 5, …",{"a":761,"b":699},"Square dot patterns: 1, 4, 9, …",{"a":763,"b":702},"Staircases of blocks: 1, 3, 6, …",{"a":765,"b":766},"Separate squares, not joined: 4, 8, 12, …","4 × n","Connect each shape pattern with the rule that counts it.","Seven shape patterns to connect to seven rules:\n\n- Matchstick squares in a row (4, 7, 10) ↔ 3 × n + 1.\n- Matchstick triangles in a row (3, 5, 7) ↔ 2 × n + 1.\n- Matchstick hexagons in a row (6, 11, 16) ↔ 5 × n + 1.\n- L-shapes of tiles (1, 3, 5) ↔ 2 × n − 1.\n- Square dot patterns (1, 4, 9) ↔ n × n.\n- Staircases (1, 3, 6) ↔ n × (n + 1) ÷ 2.\n- Separate squares that do not share sides (4, 8, 12) ↔ 4 × n.\n\nTest any rule by putting n = 1 and n = 2 and checking against the counts.",{"id":770,"type":53,"title":771,"eyebrow":772,"navLabel":773},"ch10","Predicting far-away terms, and common mistakes","Chapter 10","10 Far terms",{"id":775,"type":43,"markdown":776},"far","You now have three tools for predicting a far-away term:\n\n1. **Keep going** with the term-to-term rule. Fine for the 6th term; hopeless for the 100th.\n2. **Count the jumps** in an arithmetic sequence: nth term = first term + (n − 1) × difference.\n3. **Use a position-to-term rule** (a formula in n): put in n = 100 and calculate.\n\nFor 7, 11, 15, 19, …:\n\n- Jumps: 100th term = 7 + 99 × 4 = 7 + 396 = **403**.\n- Formula: 4 × n + 3 → 4 × 100 + 3 = **403**. The same, as it must be.\n\nYou can also **work backwards**: which term of 7, 11, 15, … is **87**? Using 4 × n + 3 = 87, we need 4 × n = 84, so n = **21**. Check: 4 × 21 + 3 = 87 ✓.",{"id":778,"type":589,"itemId":779,"prompt":780,"check":781,"hints":783,"feedback":786},"pr-far","patterns.understand-far-term","Find the **50th** term of the sequence **6, 11, 16, 21, …**",{"kind":259,"answer":782,"tolerance":446},251,[784,785],"The difference is 5. How many jumps from the 1st term to the 50th?","There are 49 jumps.",{"correct":787,"incorrect":788},"Right: 6 + 49 × 5 = 6 + 245 = 251. (Or 5 × 50 + 1 = 251.)","From the 1st to the 50th term there are 49 jumps of 5: 49 × 5 = 245. Add the first term: 6 + 245 = 251.",{"id":790,"type":589,"itemId":791,"prompt":792,"check":793,"hints":795,"feedback":798},"pr-which","patterns.understand-which-term","Matchstick squares in a row use 3 × n + 1 sticks. How many squares can you make in a row with exactly **100** matchsticks?",{"kind":259,"answer":794,"tolerance":446,"unit":528},33,[796,797],"Solve 3 × n + 1 = 100.","Take away the 1 first, then divide by 3.",{"correct":799,"incorrect":800},"Yes: 3 × n = 99, so n = 33 squares, using all 100 sticks exactly.","Set 3 × n + 1 = 100. Then 3 × n = 99 and n = 99 ÷ 3 = 33. Check: 3 × 33 + 1 = 100.",{"id":802,"type":175,"caption":803,"columns":804,"rows":807},"table-mistakes","Common mistakes with sequences",[805,361,806],"Mistake","Put it right",[808,812,816,820,824,828],[809,810,811],"Using n jumps instead of n − 1","10th term of 3, 7, 11 as 3 + 10 × 4 = 43","Only 9 jumps: 3 + 9 × 4 = 39",[813,814,815],"Using the first term as the “+ number”","Rule for 4, 7, 10 written 3 × n + 4","3 × n + 1: check n = 1 gives 4",[817,818,819],"Checking only one gap","3, 6, 12 called “add 3”","Check every gap: 6 → 12 is +6, so it is × 2",[821,822,823],"Mixing up position and term","“The 5th term of 2, 4, 6 is 5”","The 5th term is 10; 5 is the position",[825,826,827],"Forgetting shared sides","5 matchstick squares = 20 sticks","Shared sides: 3 × 5 + 1 = 16",[829,830,831],"Assuming a pattern must continue","1, 2, 4, … must be 8 next","Many rules fit 3 terms; say which rule you mean",{"id":833,"type":47,"variant":834,"title":835,"markdown":836},"model-limit","model_limit","Finite data, infinite choices","Any finite list of numbers can be continued in infinitely many ways. When a book or a test asks “what comes next?”, it really means “what comes next if the **simplest** rule you can find is the right one?”. In real life, a rule is only trustworthy if you know **why** it works (as with matchsticks, where you can see why each square adds 3). Rules that are just guessed from a few numbers can fail.",{"id":838,"type":53,"title":839,"eyebrow":840,"navLabel":841},"ch11","Check your understanding","Chapter 11","11 Check yourself",{"id":843,"type":844,"title":845,"questions":846},"quiz-understand","quiz","Sequences and rules",[847,860,873,883,896,905,914,927,940,951,964],{"itemId":848,"prompt":849,"options":850,"correct":328,"why":859},"patterns.understand-q-term","In the sequence 8, 13, 18, 23, 28, which position is the term 23 in?",[851,853,855,857],{"id":325,"label":852},"3rd",{"id":328,"label":854},"4th",{"id":331,"label":856},"5th",{"id":660,"label":858},"23rd","8 is 1st, 13 is 2nd, 18 is 3rd, 23 is 4th. The position is where a term sits, not its value.",{"itemId":861,"prompt":862,"options":863,"correct":331,"why":872},"patterns.understand-q-arith","Which sequence is arithmetic?",[864,866,868,870],{"id":325,"label":865},"2, 4, 8, 16",{"id":328,"label":867},"1, 4, 9, 16",{"id":331,"label":869},"50, 44, 38, 32",{"id":660,"label":871},"1, 1, 2, 3","Subtract 6 each time: the common difference is −6.",{"itemId":874,"prompt":875,"options":876,"correct":325,"why":882},"patterns.understand-q-geo","What is the common ratio of 5, 20, 80, 320?",[877,878,879,880],{"id":325,"label":478},{"id":328,"label":493},{"id":331,"label":490},{"id":660,"label":881},"60","20 ÷ 5 = 4, 80 ÷ 20 = 4, 320 ÷ 80 = 4.",{"itemId":884,"prompt":885,"options":886,"correct":328,"why":895},"patterns.understand-q-20th","What is the 20th term of 3, 5, 7, 9, …?",[887,889,891,893],{"id":325,"label":888},"40",{"id":328,"label":890},"41",{"id":331,"label":892},"43",{"id":660,"label":894},"39","3 + 19 × 2 = 41, or use 2 × n + 1 with n = 20.",{"itemId":897,"prompt":898,"options":899,"correct":331,"why":904},"patterns.understand-q-cube","Which number is a cube number?",[900,901,902,903],{"id":325,"label":482},{"id":328,"label":486},{"id":331,"label":487},{"id":660,"label":510},"64 = 4 × 4 × 4. (64 is also a square number, 8 × 8.) 9, 16 and 100 are squares but not cubes.",{"itemId":906,"prompt":907,"options":908,"correct":328,"why":913},"patterns.understand-q-tri","What is the 8th triangular number?",[909,910,911,912],{"id":325,"label":502},{"id":328,"label":495},{"id":331,"label":487},{"id":660,"label":508},"1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36, or 8 × 9 ÷ 2 = 36.",{"itemId":915,"prompt":916,"options":917,"correct":328,"why":926},"patterns.understand-q-fib","What comes next in 1, 1, 2, 3, 5, 8, 13, 21, …?",[918,920,922,924],{"id":325,"label":919},"29",{"id":328,"label":921},"34",{"id":331,"label":923},"42",{"id":660,"label":925},"26","Add the last two terms: 13 + 21 = 34.",{"itemId":928,"prompt":929,"options":930,"correct":331,"why":939},"patterns.understand-q-rule","Which rule gives 5, 9, 13, 17, …?",[931,933,935,937],{"id":325,"label":932},"4 × n + 5",{"id":328,"label":934},"5 × n + 4",{"id":331,"label":936},"4 × n + 1",{"id":660,"label":938},"n + 4","The step is 4, so 4 × n. For n = 1 we need 5, so add 1: 4 × n + 1.",{"itemId":941,"prompt":942,"options":943,"correct":328,"why":950},"patterns.understand-q-hex","How many matchsticks for 8 hexagons in a row (5 × n + 1)?",[944,945,946,948],{"id":325,"label":888},{"id":328,"label":890},{"id":331,"label":947},"48",{"id":660,"label":949},"46","5 × 8 + 1 = 41. Using 6 × 8 = 48 forgets the 7 shared sides.",{"itemId":952,"prompt":953,"options":954,"correct":325,"why":963},"patterns.understand-q-repunit","What is 1,111 × 1,111?",[955,957,959,961],{"id":325,"label":956},"1,234,321",{"id":328,"label":958},"123,321",{"id":331,"label":960},"1,111,111",{"id":660,"label":962},"12,321","The repunit pattern counts up to 4 and back: 1,234,321.",{"itemId":965,"prompt":966,"options":967,"correct":328,"why":973},"patterns.understand-q-alt","The pattern 20, 23, 21, 24, 22, … adds 3 then subtracts 2. What is next?",[968,969,970,972],{"id":325,"label":696},{"id":328,"label":491},{"id":331,"label":971},"20",{"id":660,"label":925},"After “− 2” (24 → 22) comes “+ 3”: 22 + 3 = 25.",{"id":975,"type":976,"prompt":977},"reflect-understand","reflection","Explain to a younger friend the difference between a **term-to-term rule** and a **position-to-term rule**, using the matchstick squares (4, 7, 10, 13, …) as your example. Which rule would you use to find the number of sticks for 1,000 squares, and why?",{"id":979,"type":980,"conceptId":981,"relation":982,"explanation":983},"conn-four-ops","connection","four-operations","helps_understand","Every row of the multiplication table is an arithmetic sequence, and geometric sequences are repeated multiplication.",{"id":985,"type":980,"conceptId":986,"relation":987,"explanation":988},"conn-properties","properties-of-numbers","related_to","Odd, even, square and triangular numbers are properties of numbers; patterns like “the sum of the first n odd numbers is n × n” connect them.",{"id":990,"type":980,"conceptId":991,"relation":987,"explanation":992},"conn-number-system","number-system","Place value is a geometric sequence: ones, tens, hundreds, thousands, each ten times the one before.",{"id":994,"type":980,"conceptId":995,"relation":982,"explanation":996},"conn-order","order-of-operations","A rule like 3 × n + 1 means multiply first, then add. Getting the order right is what makes the formula give the right term.",{"id":998,"type":999,"title":1000,"points":1001},"cheat-understand","summary","Cheat sheet",[1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012],"**Sequence:** an ordered list. **Term:** one number in it. **Position n:** where the term sits. **nth term:** the term in position n, or a formula for it.","**Term-to-term rule:** how to get the next term (“add 4”). **Position-to-term rule:** a formula in n (“4 × n + 3”). Always test a formula with n = 1.","**Difference method:** subtract neighbours. Equal differences → arithmetic. Differences growing steadily → squares or triangular numbers.","**Arithmetic:** add a constant (the common difference). nth term = first + (n − 1) × difference.","**Geometric:** multiply by a constant (the common ratio). 1, 2, 4, 8, … and 1, 10, 100, … Multiplying beats adding in the end.","**Squares:** 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. **Cubes:** 1, 8, 27, 64, 125, 216, 343, 512, 729, 1,000. **Triangular:** 1, 3, 6, 10, 15, 21, 28, 36, 45, 55.","**Fibonacci:** 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … each term is the sum of the two before. Known in India centuries before Fibonacci.","**Alternating:** the step takes turns (+3, −1, …). Split into every other term to see two simpler sequences.","**Digit patterns:** 111 × 111 = 12,321 and so on up to nine 1s; with ten 1s the carrying breaks the pattern.","**Shape rules:** matchstick squares 3n + 1, triangles 2n + 1, hexagons 5n + 1, L-shapes 2n − 1, dot squares n × n, staircases n(n + 1) ÷ 2.","**The step becomes the multiplier:** a pattern that adds 3 has a rule 3 × n + (something). Find the something from the first term.",{"id":1014,"type":1015,"sourceIds":1016},"sources-understand","sources",[1017,1018,1019,1020,1021,1022],"patterns-ncert-ganita-prakash-6","patterns-ncert-class6-algebra","patterns-mathsisfun-sequences","patterns-mathsisfun-fibonacci","patterns-wiki-fibonacci","patterns-rsos-sunflower",[1017,1018,1019,1020,1021,1022],"needs_review",{"generatedBy":1026,"notes":1027},"claude-code","Draft generated locally from a Python script with every number computed and asserted; pending owner review.","6646fe41960d3c92e90b11769f275edcea3375a1ef8eb2e5be81aabaa3da70f3",{"component:pattern-machine@1":1030,"component:sort-game@1":1031,"logic:practice":1032,"component:match-pairs@1":1033,"source:patterns-mathsisfun-fibonacci":1034,"source:patterns-mathsisfun-sequences":1035,"source:patterns-ncert-class6-algebra":1036,"source:patterns-ncert-ganita-prakash-6":1037,"source:patterns-rsos-sunflower":1038,"source:patterns-wiki-fibonacci":1039},"b3d384b49b4d138a00726767a816928da91f44fad70d7d5ccfca6db297c45839","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","b77a2fee9032c27c6a0b903d14287fc8a01076a000aba32f0ffc0e3ff54db124","1c634ad2570a389d1bc96431d0a220aa359ad82b4d33f5b55eb0b12dab8bab4c","b31773e8224c076a8a34aa4e6df704febaac5ecd4d6e1d3ba5041a31aad871e1","36726d3ff16d73627216db999b62b46580dfaf514f313774e79669de57d7c595","802a1c780b57c9da002a707e966c21f2c1663625f58a73d84a38f0551ef2d5fa","2b7f65c540dd62068d0f516bd71c5de387803d7540246de06abc95456694188a",{"state":1041,"reviewer":1042,"selfReview":1043,"reviewedAt":1044,"method":1045},"approved","The library owner",true,"2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597356]