How the Sum of Series Calculator works
A series is the sum of the terms of a sequence. This calculator handles the two most common types with exact closed-form formulas: arithmetic series, where each term increases by a constant common difference d, and geometric series, where each term is multiplied by a constant common ratio r. Instead of adding up every term one by one, these formulas let you jump straight to the total.
Arithmetic series formula
For an arithmetic series with first term a₁, common difference d, and n terms, the nth term is aₙ = a₁ + (n − 1)d. The sum of the first n terms is Sₙ = n/2 × (2a₁ + (n − 1)d), which is the same as n times the average of the first and last term, Sₙ = n/2 × (a₁ + aₙ). For example, 1 + 2 + 3 + ... + 10 has a₁ = 1, d = 1, n = 10, giving a₁₀ = 10 and S₁₀ = 10/2 × (1 + 10) = 55.
Geometric series formula
For a geometric series with first term a₁, common ratio r, and n terms, the nth term is aₙ = a₁ × r^(n − 1). The sum of the first n terms is Sₙ = a₁ × (1 − rⁿ)/(1 − r) when r ≠ 1. If r = 1, every term equals a₁, so the sum simplifies to Sₙ = a₁ × n. For example, 3 + 6 + 12 + 24 + 48 has a₁ = 3, r = 2, n = 5, giving a₅ = 48 and S₅ = 3 × (1 − 2⁵)/(1 − 2) = 93. When |r| < 1 and n is allowed to grow without bound, the sum converges to the infinite-series total S = a₁/(1 − r).
Common sources of error
- Mixing up d and r: arithmetic series use a common difference (added each step); geometric series use a common ratio (multiplied each step) — entering one where the other belongs gives a completely different total.
- Off-by-one in n: the nth term is a₁ + (n − 1)d or a₁ × r^(n − 1), not a₁ + n·d or a₁ × rⁿ — the exponent/multiplier is always n − 1 because the first term itself is already "step zero."
- Dividing by zero when r = 1: the geometric sum formula has (1 − r) in the denominator, so it is undefined at r = 1; use Sₙ = a₁ × n instead in that special case.
Checking your result
For a small arithmetic series, add the first and last term and multiply by n/2 by hand to confirm the calculator's output. For a small geometric series, add a handful of terms directly (a₁, a₁r, a₁r², ...) and compare the running total to Sₙ. If |r| > 1, the sum should grow quickly with n; if |r| < 1, later terms shrink toward zero and the sum should approach a limit.
Applications
Arithmetic series show up in evenly spaced schedules, stacking problems, and simple growth patterns (rows of seats, stacked pipes, equal pay raises each period). Geometric series appear in compound interest and loan amortization, population or investment growth models, and depreciation schedules — anywhere a quantity is repeatedly multiplied by a fixed factor.