Sum of n Terms of an AP

Core concept

For example, the sum of the first 10 terms of the AP 1,3,5,7... (a=1, d=2) is S₁₀ = 10/2 × [2(1)+(10-1)(2)] = 5 × [2+18] = 5×20 = 100. This formula is derived by pairing the first and last terms, second and second-to-last terms, and so on — each pair sums to the same total, a clever technique historically attributed to the young mathematician Carl Friedrich Gauss. This formula has practical applications in finance (calculating total savings over time) and physics (calculating total distance under constant acceleration).