IGCSE Maths / Sequences and series
The nth term of an arithmetic sequence
Edexcel International GCSE Maths (4MA1), chapter 9: Sequences and series, specification point 3.1b. Free lesson.
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An arithmetic sequence changes by the same amount each time. The rule uₙ = a + (n − 1)d gives you any term directly, including terms far along the sequence.
What the nth-term rule means
Use uₙ = a + (n − 1)d to find the value of a chosen term without listing all the terms before it. The factor n − 1 counts the gaps crossed: the first term needs no jumps, the second needs one, and so on. uₙ is the value of the term you are finding. a is the first term in the sequence. n is the term number, a whole number of 1 or more. d is the common difference, the amount added each time.
How to find and tidy the nth term
First identify the starting value a and the common difference d. Substitute them into the rule, then expand the bracket and collect the plain numbers. This gives the rule in the form dn + c. The coefficient of n is d, and the constant is a − d. For 4, 9, 14, 19, 24, the first term is 4 and the common difference is 5. A quick check is to put n = 1 into your rule. It must give the first term.
How to find a far-off term
Once you have the nth-term rule, put the requested term number in for n. For example, reaching the 50th term means crossing 49 gaps, so the rule counts 49 jumps from the first term. Use the nth-term rule to calculate a term directly, rather than writing out every term before it.
How to avoid being one term out
Using n instead of n − 1 counts one extra jump. The resulting rule can look sensible, but it starts at the second term when n = 1. Testing your answer at n = 1 catches this error. For the sequence starting at 4 with a jump of 5, 5n + 4 gives 9 at n = 1, which is the second term. The correct rule gives 4 at n = 1, so it starts in the right place.