Six consecutive integers means something like 4, 5, 6, 7, 8, and 9, where each next integer is 1 greater than the one before it.
My generally, if your first integer is n,
Then the next one is one more: n+1.
Then the next one is one more than that: n+1 + 1 or n+2.
Then the next one is one more than that: n+2 + 1 or n+3.
Then the next one is one more than that: n+3 + 1 or n+4.
Then the next one is one more than that: n+4 + 1 or n+5.
That's six consecutive integers.
And the sum of those is:
n + n+1 + n+2 + n+3 + n+4 + n+5
or
6n + 15