Let's take a look at three very familiar consecutive integers before we get more abstract with it: 1, 2, and 3. How do 2 and 3 relate to 1? Well, 2, is 1 + 1, and 3 is 1 + 2. If we look at that integers 5, 6, and 7, we see a similar pattern; 6 is 5 + 1, and 7 is 5 + 2. In general, if we want to look at any 3 consecutive integers starting with some integer [tex]n[/tex], the following integers will be [tex]n+1[/tex] and [tex]n+2[/tex].
We want to find 3 consecutive integers that add up to 54, so we'll need to add [tex]n[/tex], [tex]n+1[/tex], and [tex]n+2[/tex] together, set their sum equal to 54, and solve for [tex]n[/tex] to find them.
[tex]n+(n+1)+(n+2)=54[/tex]
Combining like terms on the left side:
[tex](n+n+n)+(1+2)=54\\ 3n+3=54[/tex]
And solving for [tex]n[/tex]:
[tex]3n+3=54\\ 3n=51\\ n=17[/tex]
The first of the three consecutive integers is 17, which means the next two must be 18 and 19. Checking our work, we find that indeed:
[tex]17+18+19=54\\
35+19=54\\
54=54[/tex]
Our three integers are 17, 18, and 19.
