Pearl writes down seven consecutive integers, and adds them up. The sum of the integers is equal to $214$ times the largest of the seven integers. What is the smallest integer that Pearl wrote down?

Answer 1

Arrange the integers in increasing order, so that the smallest integer Pearl wrote down is $n$, the next integer is $n+1$, the next integer is $n+2$, etc. Then we can express the sum of the integers as $(n)+(n+1)+(n+2)+ \cdots +(n+6)$. We know that $n+(n+1)+(n+2)+ \cdots +(n+6)=214(n+6)$. Expanding the left side gives \begin{align*}
n+n+1+n+2+\cdots +n+6&=214(n+6)\\
7n+21&=214(n+6)\\
7n+21&=214n+1284\\
-207n&=1263\\
n&=\frac{-1263}{207}=\boxed{-7}.
\end{align*}