Last updated: 2026-06-04

A geometric sequence is a list of numbers in which each term is obtained by multiplying the previous term by a fixed constant called the common ratio. The general form is:

a₁, a₁r, a₁r², a₁r³, …

where a₁ is the first term and r is the common ratio.

nth term

The nth term of a geometric sequence is:

aₙ = a₁ · rⁿ⁻¹

For example, if a₁ = 3 and r = 2, the 5th term is 3 · 2⁴ = 3 · 16 = 48.

Sum of the first n terms

The partial sum Sₙ is derived by subtracting r·Sₙ from Sₙ, which cancels all interior terms:

Sₙ = a₁ · (1 − rⁿ) / (1 − r) for r ≠ 1

When r = 1, every term equals a₁ and the sum simplifies to Sₙ = n · a₁.

Worked example. Suppose a₁ = 5 and r = 3, and you want the sum of the first 4 terms. The terms are 5, 15, 45, 135. By formula: S₄ = 5 · (1 − 3⁴) / (1 − 3) = 5 · (1 − 81) / (−2) = 5 · (−80) / (−2) = 200. Checking directly: 5 + 15 + 45 + 135 = 200. ✓

Sum to infinity

When the common ratio satisfies |r| < 1, successive terms shrink toward zero and the partial sums converge to a finite limit:

S∞ = a₁ / (1 − r)

This is obtained by letting n → ∞ in the partial-sum formula: the rⁿ term vanishes, leaving a₁ / (1 − r).

When |r| ≥ 1 the terms do not approach zero, so the series diverges — there is no finite infinite sum.

Worked example. A ball is dropped from 10 m and each bounce reaches 60 % of the previous height (r = 0.6). After the first bounce the ball rises to 6 m, then 3.6 m, then 2.16 m, and so on. The total upward distance is: 6 + 3.6 + 2.16 + … = 6 / (1 − 0.6) = 15 m.

Convergence and the ratio test

The central result is:

• |r| < 1: the series converges to S∞ = a₁ / (1 − r).
• r = 1: every term equals a₁; the partial sums grow without bound.
• r = −1: terms alternate between a₁ and −a₁; the partial sums oscillate and never settle.
• |r| > 1: terms grow in magnitude; the series diverges.

Connection to compound interest

Compound interest growth is a geometric sequence. If a principal P earns an annual rate i compounded annually, the balance after n years is:

Aₙ = P · (1 + i)ⁿ

This is a geometric sequence with a₁ = P(1 + i) and r = (1 + i). Summing regular contributions (an annuity) uses the partial-sum formula to compute the future value of a series of deposits.

Zeno's paradox

Zeno's dichotomy paradox imagines crossing a room by first travelling half the distance, then half of what remains, and so on. The total distance is the geometric series:

1/2 + 1/4 + 1/8 + … = (1/2) / (1 − 1/2) = 1

The paradox — infinitely many steps, yet a finite total — is resolved precisely by the convergence of a geometric series with |r| < 1.