Arithmetic Series Sum Calculator

Definition

An arithmetic sequence is a sequence (aₖ) with constant common difference d (Polish school texts often use r): aₖ₊₁ = aₖ + d and aₖ = a₁ + (k−1)·d. The sum of the first n terms is Sₙ = a₁ + … + aₙ. For such a sequence Sₙ = n·(a₁ + aₙ)/2. After substituting aₙ = a₁+(n−1)d you get the equivalent form Sₙ = n/2·(2a₁+(n−1)d). Intuition: end pairs share the same sum; the end average times n is the total.

Ends mode reads a₁, aₙ, n and sets d = (aₙ−a₁)/(n−1) when n > 1 (at n = 1 it takes d = 0 and requires aₙ = a₁; any other aₙ gives an error message). Step mode reads a₁, d, n, builds aₙ, then the same sum. Assumption: the entered aₙ is the nth term of the same sequence as a₁.

Historical and conceptual context

Summing equal increments is among the oldest arithmetic patterns. A popular anecdote about young Carl Friedrich Gauss (1777–1855) summing 1…100 by pairing 1+100, 2+99, … illustrates S = n·(first+last)/2. Historians note that the end-average idea was known earlier (including in Indian traditions) — the Gauss story is a teaching illustration, not a claim of priority. In high school the arithmetic series sum sits beside the nth term and before the geometric sum. A sequence is the list a₁…aₙ; a series is their sum — this page returns Sₙ.

Formal properties

Domain: a₁, aₙ, d finite reals; n an integer ≥ 1. Operations are multiplication, addition and division by 2 — no exponentiation. With d = 0: Sₙ = n·a₁. With d > 0 the sequence increases; with d < 0 it decreases. The average (a₁+aₙ)/2 is also the mean of all n terms. The two formulas agree exactly when aₙ = a₁+(n−1)d; the panel shows both numbers. Sₙ₊₁ − Sₙ = aₙ₊₁.

Full derivation

Write Sₙ = a₁ + (a₁+d) + … + (a₁+(n−1)d). Reverse and add: 2Sₙ = n·(a₁+aₙ), hence Sₙ = n·(a₁+aₙ)/2. In ends mode set d from the ends; in step mode set aₙ = a₁+(n−1)d. Compute avg = (a₁+aₙ)/2, Sₙ = n·avg and alt = n/2·(2a₁+(n−1)d). Examples: 3, 21, 7 → 84; 2, d = 3, n = 4 → 26; 20, 5, 4 → 50.

Special cases

3, 21, 7 → 84. 2, d = 3, n = 4 → 26. 20, 5, 4 → 50. 1, 10, 10 → 55. 10, d = 10, n = 4 → 100. 5, d = −2, n = 4 → 8. At n = 1 with a₁ = 7 the sum is 7. Extreme |a₁|, |d| and n may yield Infinity in floating-point arithmetic.

Numerical and computational notes

Computation: (n * (a1 + an)) / 2 and (n / 2) * (2 * a1 + (n - 1) * d) in the browser’s floating-point arithmetic. For school examples both paths agree; extremes may show floating-point rounding. n is validated by parseTermCount (integer ≥ 1). Bars show the first min(n, 8) terms; full Sₙ stays in the panel.

In ends mode the calculator does not check whether the entered aₙ “belongs” to another sequence — it assumes aₙ is the nth term of the same progression. When you know only a₁, d and n, use step mode: first aₙ = a₁ + (n − 1)·d, then the same sum Sₙ = n·(a₁ + aₙ)/2. When you know only a₁, aₙ and n, the step is d = (aₙ − a₁)/(n − 1) for n > 1. Both modes finish with the same end-average formula.

The arithmetic mean of all n terms of an arithmetic sequence equals the end average (a₁ + aₙ)/2. That is not the same page as a mean of an arbitrary list: here a constant step is part of the model. The sibling arithmetic-term page returns one aₙ; the geometric series sum multiplies by a constant ratio q instead of adding d. This page computes only the finite sum Sₙ.

Gauss pairing also works with negative d: the pairs (a₁ + aₙ), (a₂ + aₙ₋₁), … still share the same sum 2Sₙ/n. When n is odd the middle term equals the end average. At n = 1 there is no step to compute — the panel sets d = 0 and S₁ = a₁.

The term count n comes from parseTermCount: a finite whole number of at least 1. Entering 7.5 is rejected — unlike some nth-term pages, there is no floor here. Field IDs stay a, b, c and d. The result is shown without extra rounding, with no forced six-decimal trim on the primary sum.

Bars illustrate the first min(n, 8) terms aₖ = a₁ + (k−1)·d. When n > 8 the full Sₙ still sits in the result panel; the caption under the diagram states that the bars are a visual shortcut.

Related concepts and sources

Related: arithmetic nth term, geometric series sum, arithmetic mean. An infinite series with d ≠ 0 diverges. Sources: Wikipedia — Arithmetic progression; MathWorld — Arithmetic Series; textbook Sₙ = n(a₁+aₙ)/2.

In school practice, list a few terms first, then check Sₙ with the closed formula. With 3, 21 and 7 the end average is 12, and 12·7 = 84. With 2, d = 3 and n = 4 the last term is 11 and the sum is 26. With 20, 5 and 4 the step is −5 and the sum is 50. An infinite series with d ≠ 0 diverges — this page always uses finite n. The tool is educational and does not replace a textbook.