- 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**

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

- 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*

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

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

- e.g. 5, 3, 1, -1, ... is going down by 2 each term then the

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

(b)

Find a formula for the *n*^{th} term.

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

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

5 = 2×1 + *b*

5 = 2 + *b*

Now we can write the *n*^{th} term

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