Geometric Sequence & Series Calculator

Geometric Sequences Guide

Geometric Sequence Formulas

A geometric sequence multiplies by a constant ratio r each time: a, ar, ar², ar³... nth term: aₙ = a × r^(n−1). Sum of n terms: Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1, or Sₙ = na for r = 1. Sum to infinity (|r| < 1): S∞ = a/(1 − r). Example: 3, 6, 12, 24... (a = 3, r = 2). 8th term = 3 × 2⁷ = 384. Sum of 8 terms = 3(2⁸ − 1)/(2 − 1) = 3 × 255 = 765. For 0.5, 0.25, 0.125... (a = 0.5, r = 0.5): S∞ = 0.5/(1 − 0.5) = 1.

Real-World Geometric Sequences

Compound interest: principal P at rate r% per year for n years → amount = P(1 + r/100)ⁿ. This is a geometric sequence with a = P and common ratio = (1 + r/100). Radioactive decay: each half-life, activity halves → a geometric sequence with r = 0.5. Population growth (geometric): a population growing at a constant percentage rate forms a geometric sequence. Bacterial doubling: starting with 100 bacteria doubling every 30 minutes for 5 hours (10 doublings): 100 × 2¹⁰ = 102,400. The geometric seque

Sum to Infinity

The sum to infinity exists only when |r| < 1 (the series converges — terms get smaller and smaller). S∞ = a/(1 − r). For r ≥ 1: the series diverges (grows without bound). For r ≤ −1: the series oscillates and does not converge. Example: 1 + 1/2 + 1/4 + 1/8 + ... (a = 1, r = 1/2): S∞ = 1/(1 − 1/2) = 2. This means if you walked halfway across a room, then half the remaining distance, then half again... you would cover exactly 2 metres (this is Zeno's paradox resolved by geometric series).

Geometric vs Arithmetic Sequences

Arithmetic sequence: adds a constant (d) each time. aₙ = a + (n−1)d. Sum = n/2 × (2a + (n−1)d). Geometric sequence: multiplies by a constant (r) each time. aₙ = a × r^(n−1). Test: divide consecutive terms → constant = geometric. Subtract consecutive terms → constant = arithmetic. Interest: simple interest is arithmetic; compound interest is geometric — explaining why compound interest grows so much faster over long periods. Identifying which type: from 2, 6, 18, 54: ratios are 3, 3, 3 → geometri