Arithmetic & Geometric Sequence Sum Calculator

Sequences and Series Guide

Arithmetic Sequences

An arithmetic sequence is one where each term differs from the previous by a constant amount, called the common difference d. The nth term is given by uₙ = a + (n − 1)d, where a is the first term and d is the common difference. The sum of the first n terms is Sₙ = n/2 × (2a + (n − 1)d), or equivalently Sₙ = n/2 × (a + l) where l is the last term — which says the sum equals the average of the first and last terms multiplied by the number of terms, a famously elegant result attributed to Gauss as a child. A worked example: for a = 3, d = 4, the sequence is 3, 7, 11, 15, 19, ..., and u₁₀ = 3 + 9 × 4 = 39. The sum to 10 terms is S₁₀ = 10/2 × (6 + 36) = 5 × 42 = 210. To find the common difference from two terms: if u₃ = 11 and u₇ = 23, then 4d = 12, so d = 3. To find the first term: a = u₃ − 2d = 11 − 6 = 5. Arithmetic sequences appear constantly: saving a fixed amount each month, depreciating by a fixed amount per year, simple interest year by year, evenly-spaced measurements, and seating arrangements (theatre rows with increasing seats). The key recognition is constant addition — if each step adds the same number, it's arithmetic. This calculator handles arithmetic sequences with their nth term and sum formulas.

Geometric Sequences

A geometric sequence is one where each term is the previous multiplied by a constant ratio r — so growth or decay is proportional rather than additive. The nth term is uₙ = a × r^(n−1), and the sum to n terms is Sₙ = a × (1 − rⁿ) / (1 − r) for r ≠ 1. Geometric sequences explode (if r > 1) or shrink (if 0 < r < 1) far faster than arithmetic ones, which is why compound effects feel so dramatic. A worked example: for a = 2, r = 3, the sequence is 2, 6, 18, 54, 162, ..., and u₅ = 2 × 3⁴ = 162. The sum to 5 terms is S₅ = 2 × (1 − 243) / (1 − 3) = 2 × 121 = 242. For decay (0 < r < 1): for a = 100, r = 0.8, the sequence is 100, 80, 64, 51.2, ..., shrinking by 20% each step. Crucially, geometric sequences with |r| < 1 have a finite sum to infinity: S∞ = a / (1 − r). This is the formula behind many surprising results — a ball bouncing forever can travel a finite total distance, an infinite series of decreasing payments has a finite present value, and Zeno's paradoxes resolve because infinite sums can be finite. For our decay example: S∞ = 100 / (1 − 0.8) = 500. If |r| ≥ 1, the sum to infinity is infinite (or undefined for r = 1, where the sequence is just a repeated value). Geometric sequences appear in compound interest, population growth, radioactive decay, drug clearance, and many physical processes.

Applications

Sequences aren't abstract maths — they describe many real processes. Compound interest is a textbook geometric sequence: a principal P at rate r% per period grows as P, P × (1 + r/100), P × (1 + r/100)², ..., with each term r% larger than the last. After n periods the balance is P × (1 + r/100)ⁿ, which is exactly the nth term of a geometric sequence. Depreciation by a fixed percentage works the same way with r negative: a £20,000 car depreciating 15% per year has values 20,000, 17,000, 14,450, 12,283, ..., halving roughly every 4-5 years. Drug dosing creates a geometric sequence in concentration: each dose adds a fixed amount, but each existing amount also reduces by a fixed fraction each interval, so the steady-state concentration is found by summing a geometric series. Loan repayments, mortgage balances, pension growth, and the spread of infections all involve geometric sequences in their underlying maths. Arithmetic sequences are less dramatic but equally common: monthly savings into a no-interest account, depreciation by a fixed amount, evenly-spaced fence posts, and the cumulative distance covered in stages of equal length. Recognising whether a problem is arithmetic or geometric is the first step — and the key signal is whether the change between terms is additive (same amount each time, arithmetic) or multiplicative (same factor each time, geometric). This calculator handles both, working out nth terms and sums for whichever applies.

Convergence

An important question for any infinite series is whether the sum converges to a finite value or diverges to infinity. For geometric series, the rule is simple and complete: the infinite sum a + ar + ar² + ar³ + ... converges to a / (1 − r) if and only if |r| < 1, and diverges (sums to infinity) if |r| ≥ 1. This is one of the few cases in maths with a clean, total answer about infinite sums. For arithmetic sequences with d ≠ 0, the infinite sum always diverges — you keep adding constant amounts forever, so the sum grows without bound. With d = 0 the 'sequence' is just a repeated number, and the sum trivially grows linearly without bound too. Alternating series — where signs flip between + and − — can converge even when an all-positive version wouldn't: a − a/2 + a/4 − a/8 + ... = a / (1 − (−1/2)) = a / 1.5 = 2a/3 by the geometric formula with r = −1/2. The harmonic series 1 + 1/2 + 1/3 + 1/4 + ... is a famous example of a series whose terms shrink to zero but whose sum still diverges (very slowly — it takes around 12,367 terms to exceed 10, but the sum is unbounded). Convergence matters practically because it tells you whether an infinite process has a meaningful total: the total distance a bouncing ball with elastic loss travels (geometric, converges), the cumulative drug load with regular dosing and clearance (converges), and the cumulative present value of perpetual payments (converges as a perpetuity formula). This calculator flags whether your geometric series converges and gives the limit when it does.