Year: 2007
Paper: 3
Question Number: 3
Course: LFM Pure
Section: Proof by induction
No solution available for this problem.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1469.5
Banger Comparisons: 2
A sequence of numbers, $F_1, F_2, \ldots$, is defined by
$F_1=1, F_2=1$, and
\[
F_n=F_{n-1}+F_{n-2}\, \quad \text{for $n\ge 3$}.
\]
\begin{questionparts}
\item Write down the values of $F_3, F_4, \ldots , F_8$.
\item Prove that $F^{\vphantom{2}}_{2k+3}F^{\vphantom{2}}_{2k+1} -F_{2k+2}^2 = -F^{\vphantom{2}}_{2k+2}F^{\vphantom{2}}_{2k}+F_{2k+1}^2\,$.
\item Prove by induction or otherwise that
$F^{\vphantom{2}}_{2n+1}F^{\vphantom{2}}_{2n-1}-F^2_{2n}=1\,$ and deduce that $F^2_{2n}+1\,$ is divisible by $F^{\vphantom{2}}_{2n+1}\,.$
\item Prove that $F^2_{2n-1}+1\,$ is divisible by $F^{\vphantom{2}}_{2n+1}\,.$
\end{questionparts}
This question was popular. Many solutions to part (ii) were rambling and lacked a sense of direction, even if correct. The induction in (iii) was frequently incorrectly handled and a common error was to replace n by k/2. Part (iv) caused difficulties.