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Explore: Number sequences and their rules

Patterns and sequences · step 2 of 5

A question to start

A sunflower's seeds sit in spirals. Count the spirals turning one way and the spirals turning the other way, and on most sunflowers you get two numbers from this list: 21, 34, 55, 89. Where does that list come from? By the end of this page you will be able to make it yourself.

The idea

Sequences, terms and positions

A sequence is a list of numbers in order. Each number is a term. Every term has a position: 1st, 2nd, 3rd and so on.

Position 1 2 3 4 5
Term 4 7 10 13 16

Putting a sequence in a table like this makes two kinds of rule easy to see.

A "next term" rule tells you how to get from one term to the next: start at 4, add 3 each time.

A "position" rule tells you how to get a term straight from its position, without working through all the ones before it. Here: multiply the position by 3, then add 1. Check it: position 1 gives 3 × 1 + 1 = 4, position 2 gives 3 × 2 + 1 = 7, position 5 gives 3 × 5 + 1 = 16. It works for every one.

The next-term rule is easier to find. The position rule is more powerful: the 100th term is 3 × 100 + 1 = 301 straight away, with no need to write out 99 terms first.

Three famous sequences

Square numbers: 1, 4, 9, 16, 25, ... The number of dots in a square, as in the Play step. Position rule: the position times itself. The 7th square number is 7 × 7 = 49. Look at the gaps between them: 3, 5, 7, 9, ... the odd numbers. Each new square adds an L-shaped border of dots, and the border is always 2 dots longer than the last one.

Triangle numbers: 1, 3, 6, 10, 15, ... The number of dots in a triangle: a row of 1, then rows of 1 and 2, then 1, 2 and 3, and so on. Next-term rule: add 2, then 3, then 4, then 5. The 4th triangle number is 1 + 2 + 3 + 4 = 10. Bowling pins are arranged as the 4th triangle number.

Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... The rule is unusual: 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. Leonardo of Pisa, called Fibonacci, wrote about it in 1202 in a puzzle about rabbits breeding. It turns up in sunflower heads, pine cones and pineapples, because plants that add each new seed at a fixed turning angle end up packing them in spirals whose counts are Fibonacci numbers.

Worked example

Find the position rule for 2, 7, 12, 17, 22, ... and use it to find the 50th term.

  1. Make the table.

    Position 1 2 3 4 5
    Term 2 7 12 17 22

  2. Find the next-term rule: each term is 5 more than the last. So the rule will involve 5 × position.

  3. Compare 5 × position with the terms: 5, 10, 15, 20, 25. Each is 3 too big.
  4. So the position rule is 5 × position − 3. Check: 5 × 1 − 3 = 2 ✓, 5 × 4 − 3 = 17 ✓.
  5. The 50th term is 5 × 50 − 3 = 247.

The trick in step 2 always works when a sequence adds the same amount each time: the amount you add becomes the number you multiply the position by. The next step, Build, explains why.

Try it

  1. Write the next three terms of 1, 3, 6, 10, 15, ...
  2. Write the next three Fibonacci numbers after 55, 89.
  3. Find the position rule for 6, 10, 14, 18, ... and the 20th term.
  4. Is 100 a term of 3, 7, 11, 15, ...? Explain how you know.
  5. The gaps between square numbers are 3, 5, 7, 9, ... What is the gap between the 10th and the 11th square numbers?
Answers
  1. 21, 28, 36. Add 6, then 7, then 8.
  2. 144, 233, 377. 55 + 89 = 144, 89 + 144 = 233, 144 + 233 = 377.
  3. It adds 4 each time, so try 4 × position: 4, 8, 12, 16. Each is 2 too small, so the rule is 4 × position + 2. The 20th term is 4 × 20 + 2 = 82.
  4. No. The rule is 4 × position − 1, so every term is one less than a multiple of 4. 100 is a multiple of 4 itself, so it is never one less than one.
    1. The 10th square is 100 and the 11th is 121; 121 − 100 = 21. The gaps are the odd numbers, and the 10th gap (between squares 10 and 11) is 2 × 10 + 1 = 21.

Common mistake

Mixing up the position and the term. In 4, 7, 10, 13, the 3rd term is 10, but 3 is its position. When a question asks for "the 20th term", 20 is the position you put into the rule, not the answer.

Where it leads