Arithmetic & Geometric Sequence Calculator

Sequences Guide

Arithmetic Sequences

An arithmetic sequence has a constant difference between terms: 3, 7, 11, 15, 19... (d = 4). nth term: aₙ = a + (n−1)d. Sum of n terms: Sₙ = n/2 × (2a + (n−1)d) or n/2 × (first + last). Example: a=3, d=4, n=10: aₙ = 3 + 9×4 = 39. S₁₀ = 10/2 × (6 + 36) = 5 × 42 = 210. Real-world arithmetic sequences: even numbers (a=2, d=2), salary with annual increments, steps climbed on a staircase.

Geometric Sequences

A geometric sequence has a constant ratio between terms: 2, 6, 18, 54... (r = 3). nth term: aₙ = a × rⁿ⁻¹. Sum of n terms: Sₙ = a(rⁿ − 1)/(r − 1) when r ≠ 1. Example: a=2, r=3, n=5: aₙ = 2×3⁴ = 162. Real-world geometric sequences: compound interest (a=principal, r=1+rate), bacterial growth, folding paper (thickness doubles each fold), radioactive decay.

Infinite Geometric Series

A geometric series converges to a finite sum when |r| < 1. Sum to infinity: S∞ = a/(1−r). Example: 1 + 0.5 + 0.25 + 0.125 + ... (a=1, r=0.5): S∞ = 1/(1−0.5) = 2. This is why Achilles catches the tortoise — infinite geometric series can have finite sums. Diverges when |r| ≥ 1: the sum grows without bound (r=2: 1+2+4+8+... → ∞) or oscillates (r=−2).

Sequences in Finance

Compound interest is a geometric sequence: a deposit of £1,000 at 5% annual interest after n years = £1,000 × 1.05ⁿ. Mortgage repayments and annuities use the sum of a geometric series to calculate total payments. Regular savings (investing the same amount each month) form an arithmetic-then-geometric structure that underpins all retirement planning calculations.