Sum of the terms of an arithmetic progression

The sum of the first n terms of an arithmetic progression is S_n = a₁ + a₂ + ... + a_n. To find it we write the sum twice, in increasing and in reverse order, and add term by term. In a progression a₁ + a_n = a₂ + a_(n−1) = ..., because one step right adds d and one step left subtracts d. So 2S_n = (a₁ + a_n) · n, i.e. S_n = (a₁ + a_n) : 2 · n. Substituting a_n = a₁ + (n − 1)d gives S_n = (2a₁ + (n − 1)d) : 2 · n. A special case: 1 + 2 + ... + n = n(n + 1) : 2. If the number of terms n is unknown, find it from a_n = a₁ + (n − 1)d. If finding n leads to a quadratic equation, choose the natural root.