The sum of three consecutive integers is 51. What are the three numbers?

Step-by-step solution

1. 1

   Centre the variable on the middle number. Writing the three integers as $n-1,\ n,\ n+1$ instead of $n,\ n+1,\ n+2$ makes the $-1$ and $+1$ cancel, so no constant term survives:

   $(n-1)+n+(n+1)=3n$

2. 2

   Set the sum equal to 51 and solve.

   $3n=51\ \Longrightarrow\ n=17$

   The middle integer is the sum divided by the count of terms — that is, the mean of the three numbers.

3. 3

   Recover the other two integers.

   $n-1=16,\qquad n=17,\qquad n+1=18$

4. 4

   Check the sum.

   $16+17+18=51\ \checkmark$

5. 5

   Note when this trick fails. Because the middle term is the mean, three consecutive integers can only sum to a multiple of $3$. A target like $50$ has no solution in integers, since $50/3$ is not whole — a one-line feasibility test worth running before any algebra.