Sequences

Algebra and Sequences

Finding the nth term

A linear sequence changes by the same amount at every step, and that fixed step is what the nth term is built on
The nth term takes the form (step)n + (term before the first)
Find the step from the differences, then work backwards one place from the first term to get the number that follows n
A sequence that decreases has a negative step, and the number that follows n is then larger than the first term
Always test the formula on the second or third term before writing it down

Exam tip

Build the rule from the sequence itself: the number in front of n is the common difference, and the constant is the term that would sit before the first one. For 5, 11, 17 the difference is 6, the zeroth term is −1, and the nth term is 6n − 1. Substituting n = 1 back is the check.

Worked example

An increasing and a decreasing sequence

Find the nth term of 7, 11, 15, 19, … and of 20, 14, 8, 2, …

Solution:

For 7, 11, 15, 19 the differences are all 4, so the formula starts 4n
Going back one step from 7 gives 3, so the nth term is 4n + 3
Check with n = 3: 4 × 3 + 3 = 15, which matches the third term
For 20, 14, 8, 2 the differences are all −6, so the formula starts −6n
Going back one step from 20 gives 26, so the nth term is 26 − 6n
Check with n = 4: 26 − 24 = 2, which matches the fourth term

Testing whether a number belongs

Set the nth term equal to the value in question and solve the equation for n
A whole-number answer means the value is in the sequence, and n tells you which term it is
A fraction or a decimal answer means the value never appears in the list, and saying so is what earns the mark
For the sequence with nth term 4n + 3, solving 4n + 3 = 95 gives n = 23, so 95 is the 23rd term
Solving 4n + 3 = 100 gives n = 24.25, which is not a position, so 100 never appears