The sum of 6 consecutive integers is 9. What is the lowest integer?

Concept

A run of consecutive integers is an arithmetic sequence with common difference 1. If the smallest term is n, then k consecutive integers are n, n+1, …, n+(k−1), and their sum equals k·n + (0+1+…+(k−1)) = k·n + k(k−1)/2.

Application

Here k = 6 and the sum is 9. Substitute into the formula and solve for the lowest integer n:

1. Write the sum of the 6 terms: n + (n+1) + (n+2) + (n+3) + (n+4) + (n+5).

2. Combine like terms: 6n + (1+2+3+4+5) = 6n + 15.

3. Set equal to the given sum: 6n + 15 = 9.

4. Subtract 15 from both sides: 6n = −6.

5. Divide both sides by 6: n = −1.

Cross-check

With n = −1 the six integers are −1, 0, 1, 2, 3, 4, and their sum is −1 + 0 + 1 + 2 + 3 + 4 = 9, which matches the given total. So the lowest integer is −1.