Sum of First n Integers

This formula gives the total when you add the counting numbers from 1 up to n without writing them all out. It works because the list can be paired from the ends: 1 with n, 2 with n-1, and so on, and each pair has the same sum, n+1. There are n/2 such pairs, so the total is n(n+1)/2.

Use this when you need the sum of a consecutive run of whole numbers, such as counting objects in rows, finding the total of the first n terms of a simple counting pattern, or checking arithmetic-series sums quickly. It is especially useful in combinatorics, number theory, and mental math.

• Using n instead of n+1 in the formula
• Forgetting that the formula applies to consecutive integers starting at 1
• Multiplying by n/2 incorrectly when n is odd without understanding that the final formula still gives an integer