Arithmetic and Geometric Sequences and Series
An arithmetic sequence has a constant difference d between consecutive terms, so its nth term is a + (n - 1)d, where a is the first term. A geometric sequence has a constant ratio r, so its nth term is ar raised to the power (n - 1). The corresponding series are their sums: for an arithmetic series the sum of n terms is (n/2) times (2a + (n - 1)d), which can be read as the number of terms times the average of the first and last; for a geometric series it is a(1 - r to the power n) divided by (1 - r). Recognising which type a sequence is comes first, and it is settled by asking whether consecutive terms differ by adding the same amount or by multiplying by the same amount.
Before you start
No specific prerequisites - this is a good place to start.
Method
- Decide the type: subtract consecutive terms, and if the difference is constant it is arithmetic; divide consecutive terms, and if the ratio is constant it is geometric.
- Extract a and d, or a and r, before substituting anything, and write them down explicitly.
- For a particular term, substitute n into the nth term formula, taking care that the exponent and the bracket use (n - 1), not n.
- For a sum, choose the matching series formula and substitute the same a, d or r, and the number of terms n. A common error is using the last term's position when the count of terms is different.
- When two terms are given but not the first, form two equations and solve them simultaneously: for arithmetic, subtracting eliminates a; for geometric, dividing eliminates a.
- Sense check the answer: an arithmetic sequence with a positive difference grows steadily, while a geometric sequence with a ratio greater than 1 grows very fast, so a sum that looks far too small usually means the wrong formula was used.
Worked example
An arithmetic sequence has first term 5 and common difference 3. (a) Find the 20th term. (b) Find the sum of the first 20 terms. A geometric sequence has first term 3 and common ratio 2. (c) Find the sum of its first 8 terms.
- For the 20th term use a + (n - 1)d with a = 5, d = 3 and n = 20: 5 + 19 times 3.
- Evaluate: 19 times 3 = 57, so the 20th term is 5 + 57 = 62.
- For the sum use (n/2)(2a + (n - 1)d) = (20/2)(2 times 5 + 19 times 3) = 10 times (10 + 57).
- Evaluate: 10 times 67 = 670, so the sum of the first 20 terms is 670.
- For the geometric series use a(r to the power n - 1) divided by (r - 1) since r is greater than 1: 3(2 to the power 8 - 1) divided by (2 - 1).
- Evaluate: 2 to the power 8 = 256, so the sum is 3 times 255 divided by 1 = 765.
Practice questions
Try each question, then tap to reveal the answer.
Q1Is 2, 6, 18, 54 arithmetic or geometric, and what is the key value?Show answer
Answer: Geometric, with common ratio 3.
Q2Find the nth term of the arithmetic sequence 7, 11, 15, 19.Show answer
Answer: a = 7 and d = 4, so the nth term is 7 + 4(n - 1) = 4n + 3.
Q3Find the 10th term of a geometric sequence with first term 5 and ratio 2.Show answer
Answer: 5 times 2 to the power 9 = 5 times 512 = 2560.
Q4Find the sum of the first 12 terms of the arithmetic sequence with a = 4 and d = 6.Show answer
Answer: (12/2)(2 times 4 + 11 times 6) = 6 times (8 + 66) = 6 times 74 = 444.
Q5An arithmetic sequence has 3rd term 11 and 7th term 27. Find d.Show answer
Answer: Four steps take the term from 11 to 27, so 4d = 16 and d = 4.
Q6Why does subtracting two equations eliminate a in an arithmetic sequence problem?Show answer
Answer: Both equations contain a exactly once with the same coefficient, so subtraction removes it and leaves an equation in d alone.
Q7What is the sum of the first 5 terms of the geometric sequence 2, 6, 18, 54, 162?Show answer
Answer: 2 + 6 + 18 + 54 + 162 = 242, which matches 2(3 to the power 5 - 1) divided by 2 = 2 times 242/2.
Exam-style questions
Written in the style of a IGCSE Maths exam paper, with a full mark scheme.
The 4th term of an arithmetic sequence is 23 and the 9th term is 48. (a) Find the first term and the common difference. (b) Find the sum of the first 30 terms.
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A geometric sequence has second term 12 and fifth term 96. Find the first term and the common ratio, showing your method.
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Free printable worksheet
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