Geometric Series Sum (First n Terms)
Sum the first n terms of a geometric sequence from its first term and common ratio, with the standard closed-form formula.
How it works
Sum = a₁ × (rⁿ − 1) ÷ (r − 1) for r ≠ 1 · Sum = a₁ × n for r = 1 · Sum(∞) = a₁ ÷ (1 − r) when |r| < 1
A geometric series adds terms that each multiply by a fixed ratio r. The closed form a₁(rⁿ − 1)/(r − 1) comes from subtracting r times the series from the series itself, which cancels every middle term. When |r| < 1 the terms shrink toward zero and even an infinite number of them add to a finite total, a₁/(1 − r) — the basis of many limit and finance results.
Worked example
For a₁ = 5, r = 2, n = 8: the sum is 5 × (2⁸ − 1) ÷ (2 − 1) = 5 × 255 = 1,275. The last term is 5 × 2⁷ = 640.
Sources: Geometric series — Wikipedia
Frequently asked questions
What is the sum-of-geometric-series formula?
For a common ratio r ≠ 1, Sum = a₁ × (rⁿ − 1) / (r − 1). If r = 1 every term equals a₁, so the sum is simply a₁ × n. This calculator handles both cases automatically.
When does an infinite geometric series have a sum?
Only when |r| < 1, so the terms shrink to zero. Then the infinite sum equals a₁ / (1 − r). If |r| ≥ 1 the terms do not vanish and the infinite series grows without bound.
What if the ratio is exactly 1?
Then every term is the same as the first, and the formula would divide by zero. The correct sum is a₁ × n, which the calculator uses whenever r equals 1.
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