Year: 2008
Paper: 3
Question Number: 8
Course: UFM Additional Further Pure
Section: Sequences and Series
No solution available for this problem.
Most candidates attempted five, six or seven questions, and scored the majority of their total score on their best three or four. Those attempting seven or more tended not to do well, pursuing no single solution far enough to earn substantial marks.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
\begin{questionparts}
\item
The coefficients in the series
\[
S= \tfrac13 x + \tfrac 16 x^2 + \tfrac1{12} x^3 + \cdots + a_rx^r +
\cdots
\]
satisfy a recurrence relation of the form $a_{r+1} + p a_r =0$. Write
down the value of $p$.
By considering $(1+px)S$, find an expression for the sum to infinity
of $S$ (assuming that it exists). Find also
an expression for the sum of the first $n+1$ terms of $S$.
\item
The coefficients in the series
\[
T=2 + 8x +18x^2+37 x^3 +\cdots + a_rx^r + \cdots
\]
satisfy a recurrence relation of the form $a_{r+2}+pa_{r+1} +qa_r=0$.
Find an expression for the sum to infinity of $T$ (assuming that it
exists). By expressing $T$ in partial fractions, or
otherwise,
find
an expression for the sum of the first $n+1$ terms of $T$.
\end{questionparts}
Three fifths attempted this with most scoring about two thirds of the marks. Apart from minor errors, the last part (expressing T in partial fractions etc.) was the pitfall for most.