Sum of the terms of an arithmetic sequence

The formula for the sum

To add the consecutive terms of an arithmetic sequence, we use: S = (number of terms) * (first term + last term) / 2

In other words, for the sum of u_0 to u_n (i.e. n+1 terms): S = (n+1) * (u_0 + u_n) / 2

Gauss’s trick

This formula stems from a trick attributed to Gauss: add the sequence to itself written in reverse order. Each pair (first + last, second + penultimate, …) always gives the same sum.

Example: 1 + 2 + … + 100. S = 100 * (1 + 100) / 2 = 100 * 101 / 2 = 5050

Example with a general arithmetic sequence

Let’s calculate the sum of the first 10 terms of u_n = 3 + 5n (u_0 = 3, r = 5). u_9 = 3 + 5 × 9 = 48 S = 10 × (3 + 48) / 2 = 10 × 51 / 2 = 255

Terms	Number	Sum
u_0 to u_9	10	255

Pitfalls to avoid

• Count the number of terms carefully: from u_0 to u_n, there are (n+1) terms, not n.
• Don’t forget to divide by 2: this is the most common mistake in this calculation.
• Check what the last term of the required sum actually is before applying the formula, especially if the sequence does not start at u_0.