IGCSE Maths / Sequences and series
Using the nth term formula in reverse
Edexcel International GCSE Maths (4MA1), chapter 9: Sequences and series, specification point 3.1b. Free lesson.
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Use the nth term rule backwards by setting it equal to a given number and solving for n. This tells you whether the number is in the sequence, where it appears, or which term first passes a limit.
Find the nth term rule first
For an arithmetic sequence, the nth term rule gives the value at position n. In the sequence 4, 11, 18, 25, each term increases by 7. The first term is 4, so the rule is 4 + (n − 1) × 7, which simplifies to 7n − 3. The first term is 4 and the common difference is 7. The rule is uₙ = 7n − 3.
Work out which term a number is
To find the position of a number, set the nth term rule equal to that number and solve for n. A whole-number value of n means the number occurs in the sequence. Here, n gives the position, not the value of the term.
Show when a number is not in the sequence
A sequence only has terms at whole-number positions. If solving for n gives a fraction, the number falls between two positions, so it cannot be a term. State that n is not a whole number to explain why the number is not in the sequence. For 100, solving 7n − 3 = 100 gives n = 103/7. Since 103/7 is between 14 and 15, 100 is not a term.
Find the first term past a limit
To find the first term greater than a limit, first solve as if a term equalled the limit. The resulting n may not be a whole number. Round n up to the next whole-number position, then substitute that position into the rule to find the term itself. If the question asks for the term, give its value, not just its position.