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.

Sum of Arithmetic Series

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

Result

S100

5,050.00

Avg term 50.5000; last term 100.00.

First term a₁1
Common diff d1
Number of terms100
Last term aₙ100.00
Sum5,050.00
Average term50.5000

Step-by-step

aₙ = a₁ + (n−1)d = 1 + 99 × 1 = 100.00.
Sₙ = n/2 × (a₁ + aₙ) = 100/2 × (1 + 100.00) = 5,050.00.

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.