Linear Sequences

What is a linear sequence?

A linear sequence is one where the terms go up (or down) by the same amount each time
- eg 1, 4, 7, 10, 13, … (add 3 to get the next term)
- 15, 10, 5, 0, -5, … (subtract 5 to get the next term)
A linear sequence is often referred to as an arithmetic sequence
If we look at the differences between the terms, we see that they are constant

What should we be able do with linear sequences?

You should be able to recognise and continue a linear sequence
You should also be able to find a formula for the n^thterm of a linear sequence in terms of n
This formula will be in the form:
n^th term = dn + b, where;
- d is the common difference
- b is a constant that makes the first term “work”

How do I find the n^th term of a linear (arithmetic) sequence?

Find the common difference between the terms – this is d
Put the first term and n = 1 into the formula, then solve to find b

Exam Tip

If a sequence is going up by d each time, then its n^th term contains dn
- e.g. 5, 7, 9, 11, is going up by 2 each term so the n^th term contains 2n
- (the complete n^th term for this example is 2n + 3)
If a sequence is going down by d each time, then its n^th term contains −dn
- e.g. 5, 3, 1, -1, ... is going down by 2 each term then the n^th term contains −2n
- (the complete n^th term for this example is −2n + 7)

Worked example

Given the sequence 5, 7, 9, 11, 13, ...

(a)

Find the next three terms.

Looking at the difference between the terms, we see that they are all 2. So this is a linear sequence with common difference 2

So the next three terms are

13 + 2 = 15

15 + 2 = 17

17 + 2 = 19

15, 17, 19

(b)

Find a formula for the n^th term.

In part (a) we established that the common difference is 2. So d = 2

n^thterm = 2n + b

The first term is 5. Substitute this and n = 1 into the formula, and solve for b

5 = 2×1 + b

5 = 2 + b

b = 3

Now we can write the n^th term

2n + 3

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