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?
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*}