Rules, terms and sequences

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?

This 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.

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”.

Here 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.

A 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.

There are two ways to describe the rule of a sequence.

A 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.

A 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.

The 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”.

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.

What the differences tell you:

1. All the same → the sequence adds a fixed amount: an arithmetic sequence. The rule is “add the difference”.
2. They go up by the same amount (like 3, 5, 7, 9) → a square-number-like sequence. Look for n × n hidden inside it.
3. 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.
4. They take turns (like +3, −1, +3, −1) → an alternating pattern.
5. Each term is the sum of the two before → a Fibonacci-type sequence.

If none of these fit, try the ratio (divide each term by the one before), look at the digits, or think about pictures.

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.

• 4, 9, 14, 19, 24, … common difference +5
• 100, 93, 86, 79, … common difference −7 (a decreasing arithmetic sequence)
• 0.5, 1, 1.5, 2, … common difference +0.5
• −6, −2, 2, 6, 10, … common difference +4, starting below zero

Arithmetic 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.

To find a far-away term, count the jumps. From the 1st term to the nth term there are n − 1 jumps. So:

nth term = first term + (n − 1) × common difference

In a geometric sequence each term is the one before multiplied by the same number, the common ratio.

• 1, 2, 4, 8, 16, 32, … ratio 2 (the powers of 2)
• 1, 3, 9, 27, 81, … ratio 3 (the powers of 3)
• 1, 10, 100, 1,000, 10,000, … ratio 10: this is our place-value system! Ones, tens, hundreds, thousands.
• 64, 32, 16, 8, 4, … ratio ½ (halving)
• 81, 27, 9, 3, 1, … ratio ⅓

Geometric 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.

To 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).

Some sequences are so useful that they have their own names. You should know the first ten or so of each by heart.

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.

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.

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).

Here is a sequence with a different kind of rule: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

Each 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.

The 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).

Say 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.

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.

Alternating 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.

Another kind of alternating pattern flips the sign: 1, −1, 1, −1, … or 1, −2, 3, −4, 5, …

The multiplication table (the 10 by 10 grid of times tables) is full of patterns:

• 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.
• That diagonal holds the square numbers: 1, 4, 9, 16, …, 100.
• The 5s row ends in 5, 0, 5, 0, … and the 10s row ends in 0.
• The 2s, 4s, 6s, 8s and 10s rows contain only even numbers.
• Every row is an arithmetic sequence: the 7s row adds 7.
• In any row, the last digits repeat: the 4s row ends in 4, 8, 2, 6, 0, 4, 8, 2, 6, 0.

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.

Take matchstick squares in a row: 4, 7, 10, 13, … sticks. Two ways to see the rule:

• Term-to-term: each new square needs 3 more sticks (top, bottom, and one side). Add 3.
• 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.

Check: n = 1 gives 4 ✓; n = 4 gives 13 ✓. So 50 squares need 3 × 50 + 1 = 151 sticks.

You now have three tools for predicting a far-away term:

1. Keep going with the term-to-term rule. Fine for the 6th term; hopeless for the 100th.
2. Count the jumps in an arithmetic sequence: nth term = first term + (n − 1) × difference.
3. Use a position-to-term rule (a formula in n): put in n = 100 and calculate.

For 7, 11, 15, 19, …:

• Jumps: 100th term = 7 + 99 × 4 = 7 + 396 = 403.
• Formula: 4 × n + 3 → 4 × 100 + 3 = 403. The same, as it must be.

You 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 ✓.