Consecutive Integer Sum
The sum of five consecutive integers is equal to the sum of the next three consecutive integers. What is the largest of these eight integers?
Answer
Let the five consecutive integers be \(n, n+1, n+2, n+3, n+4\). Their sum is
\[
5n + 10.
\]
The next three consecutive integers are \(n+5, n+6, n+7\) with a sum of
\[
3n + 18.
\]
Setting the sums equal:
\[
5n + 10 \;=\; 3n + 18.
\]
Subtract \(3n\) from both sides:
\[
2n + 10 \;=\; 18.
\]
Subtract 10:
\[
2n \;=\; 8
\quad\Longrightarrow\quad
n \;=\; 4.
\]
Thus, the eight consecutive integers are
\[
4,\;5,\;6,\;7,\;8,\;9,\;10,\;11,
\]
whereby the largest of these eight integers is
\[
\boxed{11}.
\]