Sequences and Series: Arithmetic vs Geometric in Plain English
May 9, 2026 · 5 min · sequences and series · arithmetic sequence · geometric sequence
Sequences and series are pattern questions in disguise. You only need to spot whether it's arithmetic (adding) or geometric (multiplying).
Arithmetic sequences
Each term is the previous term plus a constant d (the common difference).
- nth term: a + (n-1)d
- Sum of first n terms: n/2 × (2a + (n-1)d)
Where a is the first term, n is how many terms.
Geometric sequences
Each term is the previous term times a constant r (the common ratio).
- nth term: a × r^(n-1)
- Sum of first n terms: a × (1 - rⁿ) / (1 - r), for r ≠ 1
- Sum to infinity (only if |r| < 1): a / (1 - r)
Spot which is which
Look at consecutive differences and consecutive ratios.
- 2, 5, 8, 11, ... → differences are 3, 3, 3 (arithmetic, d = 3)
- 2, 6, 18, 54, ... → ratios are 3, 3, 3 (geometric, r = 3)
- 2, 4, 8, 16, ... → ratios are 2, 2, 2 (geometric, r = 2)
If neither differences nor ratios are constant, it's neither — and you'll need a different formula.
Common pitfalls
- Mixing up nth term and sum
- Forgetting the (n - 1) in the arithmetic formula
- Using the sum-to-infinity formula when |r| ≥ 1 (it doesn't converge)
- Off-by-one errors in counting how many terms
A worked example
Find the 10th term and the sum of the first 10 terms of 3, 7, 11, 15, ...
This is arithmetic: a = 3, d = 4.
- 10th term: 3 + 9 × 4 = 39
- Sum: 10/2 × (6 + 9 × 4) = 5 × 42 = 210