Let $\frac{P_{n}}{Q_{n}}=[\underbrace{1 ; 1, \ldots, 1}_{n}]$. What are $P_{n}$ and $Q_{n}$?
Note that $\frac{P_{n+1}}{Q_{n+1}}=1+\frac{1}{1+\frac{P_{n}}{Q_{n}}}=\frac{Q_{n}}{P_{n}+Q_{n}}$, that is, $P_{n+1}=Q_{n}, Q_{n+1}=P_{n}+Q_{n}$. Therefore, $P_{n+2}=Q_{n+1}=$ $P_{n}+Q_{n}=P_{n}+P_{n+1}$. Since

$P_{1 / Q_{1}}=1, P_{2 / Q_{2}}=2$, then $P_{1}=1, P_{2}=2$, and the sequence $P_{n}$ coincides with the sequence $F_{n+1}$ of Fibonacci numbers.

## Answer

$P_{n}=F_{n+1}, Q_{n}=F_{n}$.

Send a comment