IGCSE Maths / Sequences and series
Arithmetic sequences, a and d
Edexcel International GCSE Maths (4MA1), chapter 9: Sequences and series, specification point 3.1a. Free lesson.
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An arithmetic sequence has the same difference between each pair of neighbouring terms. Find its first term, a, and common difference, d, then use them to write a rule for the nth term.
What makes a sequence arithmetic
A term is one number in the list. The common difference, d, is the amount added to each term to get the next one. It must be the same each time. A pattern by itself is not enough: if the gaps are different, the sequence is not arithmetic and the arithmetic sequence rule does not apply. For 2, 6, 18, 54, the gaps are 4, 12 and 36. They are not equal, so this is not arithmetic. For 5, 9, 13, 17, every gap is 4, so the sequence is arithmetic and d = 4.
How to find a and d
The first term is a. Find d by subtracting one term from the term immediately after it. Check at least two gaps agree before deciding the sequence is arithmetic. For 7, 11, 15, 19, 23, the first term is a = 7. The gaps are all +4, so d = 4.
How to write the nth-term rule
Use uₙ = a + (n − 1)d. Here, uₙ means the term in position n. The number of jumps to reach position n is n − 1, because the first term is already at position 1. This rule is not printed on the formula sheet, so learn it. Substitute a and d into the rule, keeping brackets around n − 1. To reach the fifth term, you make four jumps, not five.
How to handle a sequence that falls
A falling arithmetic sequence has a negative common difference. The minus sign is part of d, so it must stay in the substitution and working. For 23, 20, 17, 14, 11, each gap is −3. So a = 23 and d = −3.
How to check the common difference
One gap suggests a value for d, but checking another gap confirms the sequence is arithmetic. Subtract each term from the one after it, in that order. For 12, 19, 26, 33: 19 − 12 = 7 and 26 − 19 = 7. The gaps agree, so d = 7.