Sequences

Algebra and Sequences

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Look at the gaps first

Write the differences between consecutive terms underneath the sequence
Equal differences mean the sequence goes up or down in fixed steps, so add the same amount again
Differences that themselves form a pattern still tell you what comes next
- In 5, 6, 8, 12, 20 the differences are 1, 2, 4 and 8, doubling at every step; one more doubling gives 16, and 20 + 16 = 36
Where the differences give nothing, test whether each term is being multiplied by the same number instead

Number sequences worth recognising

Square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
Cube numbers: 1, 8, 27, 64, 125, 216
Triangular numbers: 1, 3, 6, 10, 15, 21, formed by adding 1, then 2, then 3, and so on
Prime numbers: 2, 3, 5, 7, 11, 13, which follow no difference pattern at all
Recognising these on sight turns many sequence questions into a one-line comparison

Exam tip

Check the difference is the same all the way along before deciding it is linear: two terms agreeing proves nothing. If the differences are not constant, take the differences of the differences, and if those are not constant either, test whether each term is being multiplied by a fixed number instead.

Worked example

Recognising a familiar sequence

Write down the next two terms of each sequence: (a) 1, 4, 9, 16 (b) 2, 4, 8, 16 (c) 1, 3, 6, 10

Solution:

(a) The gaps are 3, 5, 7 — not constant, so this is not linear
These are the square numbers 1², 2², 3², 4²
The next two are 5² and 6², giving 25 and 36
(b) Each term is double the one before, so the rule is multiplicative rather than additive
Doubling 16 gives 32, and doubling again gives 64, so the next two are 32 and 64
(c) The gaps are 2, 3, 4, increasing by one each time
These are the triangular numbers, and the next gaps are 5 and 6
The next two terms are 15 and 21
Check the gaps first every time: a constant gap means linear, a constant multiplier means exponential