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Linear Sequences (AQA GCSE Maths)
Revision Note
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 nth term of a linear sequence in terms of n
- This formula will be in the form:
nth term = dn + b, where;
-
- d is the common difference
- b is a constant that makes the first term “work”
How do I find the nth 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
Examiner Tip
- If a sequence is going up by d each time, then its nth term contains dn
- e.g. 5, 7, 9, 11, is going up by 2 each term so the nth term contains 2n
- (the complete nth term for this example is 2n + 3)
- If a sequence is going down by d each time, then its nth term contains −dn
- e.g. 5, 3, 1, -1, ... is going down by 2 each term then the nth term contains −2n
- (the complete nth 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 nth term.
In part (a) we established that the common difference is 2. So d = 2
nth term = 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 nth term
2n + 3
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