Year: 2012
Paper: 3
Question Number: 8
Course: UFM Pure
Section: Sequences and series, recurrence and convergence
No solution available for this problem.
The number of candidates attempting more than six questions was, as last year, about 25%, though most of these extra attempts achieved little credit.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
The sequence $F_0$, $F_1$, $F_2$, $\ldots\,$ is defined
by $F_0=0$, $F_1=1$ and, for $n\ge0$,
\[
F_{n+2} = F_{n+1} + F_n
\,.
\]
\begin{questionparts}
\item Show that $F_0F_3-F_1F_2 = F_2F_5- F_3F_4\,$.
\item
Find the values of $F_nF_{n+3} - F_{n+1}F_{n+2}$
in the two cases that arise.
\item
Prove that, for $r=1$, $2$, $3$, $\ldots\,$,
\[
\arctan \left( \frac 1{F_{2r}}\right)
=\arctan \left( \frac 1{F_{2r+1}}\right)+
\arctan \left( \frac 1{F_{2r+2}}\right)
\]
and hence evaluate the following sum (which you may assume converges):
\[
\sum_{r=1}^\infty \arctan \left( \frac 1{F_{2r+1}}\right)
\,.
\]
\end{questionparts}
This was the most popular question attempted by over 83% of candidates, and the third most successful with, on average, half marks being scored. Part (i) caused no problems, though some chose to obtain the result algebraically. Part (ii) was not well attempted, with a number stating the two values the expression can take but failing to do anything else or failing with the algebra. Part (iii) was generally fairly well done although frequently the details were not quite tied up fully.