B5. In Sally's sequence, every term after the second is equal to the sum of the previous two terms. Also, every term is a positive integer. Her eighth term is 400 .
Find the minimum value of the third term in Sally's sequence.
SolUTION
Let the first two terms of Sally's sequence be $m$ and $n$ respectively. Then the next six terms of the sequence are $m+n, m+2 n, 2 m+3 n, 3 m+5 n, 5 m+8 n$ and $8 m+13 n$. So $8 m+13 n=400$. Note that 400 is a multiple of 8 .
Therefore, as 8 and 13 are coprime, $n$ is a multiple of 8 . Also, $n<\frac{400}{13}<31$.
The table shows possible values of $n$ and the corresponding values of $m$ and the third term in the sequence, $m+n$.
\begin{tabular}{crc}
$m$ & $n$ & $m+n$ \\
\hline 37 & 8 & 45 \\
24 & 16 & 40 \\
11 & 24 & 35
\end{tabular}

As can be seen, the minimum value of the third term in Sally's sequence is 35 .