Let $f_1$, $f_2$, $f_3$, $\dots$ be a sequence of numbers such that
\[
  f_n = f_{n - 1} + f_{n - 2}
\]for every integer $n \ge 3$.  If $f_7 = 83$, what is the sum of the first 10 terms of the sequence?
Let's express each of the first 10 terms using only $f_1$ and $f_2$:
\begin{align*}
f_1 &= f_1, \\
f_2 &= f_2, \\
f_3 &= f_1 + f_2, \\
f_4 &= f_1 + 2f_2, \\
f_5 &= 2f_1 + 3f_2, \\
f_6 &= 3f_1 + 5f_2, \\
f_7 &= 5f_1 + 8f_2, \\
f_8 &= 8f_1 + 13f_2, \\
f_9 &=13f_1 + 21f_2, \\
f_{10} &= 21f_1 + 34f_2.
\end{align*}(Notice anything interesting about the coefficients? These are the Fibonacci numbers!)

Adding both sides tells us that the sum of the first 10 terms is
$$55f_1+88f_2 = 11(5f_1+8f_2) = 11f_7 = 11\cdot83 = \boxed{913} .$$