Arithmetic & geometric sequence calculator (AP & GP)
Find the nth term and the sum of an arithmetic (AP) or geometric (GP) sequence.
How the calculation works
In an AP you add the ratio each term: aₙ = a₁ + (n−1)·r and Sₙ = n·(a₁+aₙ)/2. In a GP you multiply by the ratio: aₙ = a₁·rn−1 and Sₙ = a₁·(rⁿ−1)/(r−1).
Examples
- AP 2, 5, 8… (ratio 3): 10th term = 29; sum of first 10 = 155.
- GP 2, 6, 18… (ratio 3): 5th term = 2×3⁴ = 162.
Frequently asked questions
What is the difference between AP and GP?
In an AP each term adds a constant (2, 5, 8, 11…). In a GP each term multiplies by a constant (2, 6, 18, 54…).
How do I find the ratio?
In an AP, subtract neighbors (5 − 2 = 3). In a GP, divide (6 ÷ 2 = 3).
Is there an infinite GP sum?
Yes, when |ratio| < 1: S = a₁ ÷ (1 − ratio). The terms shrink and the sum converges.
What is the difference between arithmetic and geometric progressions?
An arithmetic progression adds a constant; a geometric one multiplies. Starting at 1 with ratio 2, the 30th terms are 59 and over 536 million.
How do I sum 1 to 100 quickly?
Pair the ends: each pair sums to 101 and there are 50 pairs, giving 5,050.
Can an infinite sum be finite?
Yes, when the ratio is between −1 and 1. Summing 1/2 + 1/4 + 1/8 + … forever gives exactly 1.
Where do progressions show up in real life?
Arithmetic in simple interest and linear depreciation; geometric in compound interest, population growth and drug half-life.
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Updated on June 30, 2026 · by Rafael Rossi · Methodology & sources