Sum of Arithmetic Series

Sₙ = n × (a₁ + aₙ) / 2 = n/2 × (2a₁ + (n−1)d).

Inputs

Result

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General calculation reads

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How to use this calculator

  • Enter a₁, d, n.

About this calculator

Sum of arithmetic series formula. Famous example: Gauss's schoolboy sum 1+2+...+100 = 100/2 × 101 = 5050. The formula avoids tedious term-by-term addition for any arithmetic sequence. Useful in counting (sum of natural numbers, sum of odd numbers = n², triangle numbers), staircase/seating problems, simple-interest schedules.

Frequently asked

Gauss anecdote?+
Age 7: asked to sum 1-100. Noticed pairs (1+100, 2+99...) all = 101, 50 pairs → 5050. The formula in essence.
Sum of first n natural numbers?+
1+2+...+n = n(n+1)/2. Special case a₁=1, d=1: S = n/2 × (1 + n) = n(n+1)/2.
Sum of first n odd numbers?+
1+3+5+...+(2n−1) = n². a₁=1, d=2.
Sum of first n even numbers?+
2+4+6+...+2n = n(n+1). a₁=2, d=2.
Negative d (decreasing)?+
Works the same. Sum can be positive or negative depending on terms.

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