Year: 2010
Paper: 3
Question Number: 1
Course: LFM Pure and Mechanics
Section: Arithmetic and Geometric sequences
No solution available for this problem.
About 80% of candidates attempted at least five questions, and well less than 20% made genuine attempts at more than six. Those attempting more than six questions fell into three camps which were those weak candidates who made very little progress on any question, those with four or five fair solutions casting about for a sixth, and those strong candidates that either attempted 7th or even 8th questions as an "insurance policy" against a solution that seemed strong but wasn't, or else for entertainment!
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1500.8
Banger Comparisons: 2
Let $x_{\low1}$, $x_{\low2}$, \ldots, $x_n$ and
$x_{\vphantom {\dot A} n+1}$ be any fixed real numbers.
The numbers $A$ and $B$ are defined by
\[
A = \frac 1 n
\sum_{k=1}^n x_{ \low k}
\,, \ \ \
B= \frac 1 n
\sum_{k=1}^n (x_{\low k}-A)^2
\,, \ \ \
\]
and the numbers $C$ and $D$ are defined by
\[
C = \frac 1 {n+1}
\sum\limits_{k=1}^{n+1} x_{\low k}
\,,
\ \ \
D = \frac1{n+1}
\sum_{k=1}^{n+1} (x_{\low k}-C)^2
\,.
\]
\begin{questionparts}
\item
Express $ C$ in terms of $A$, $x_{\low n+1}$ and $n$.
\item Show that $ \displaystyle
B= \frac 1 n
\sum_{k=1}^n x_{\low k}^2
- A^2\,$.
\item Express $D $ in terms of $B$, $A$, $x_{\low n+1}$ and $n$.
Hence show that $(n + 1)D \ge nB$
for all values of $x_{\low n+1}$, but that $D < B$
if and only if
\[
A-\sqrt{\frac{(n+1)B}{n}} < x_{\low n+1} <
A+\sqrt{\frac{(n+1)B}{n}}\,.
\]
\end{questionparts}
This was a very popular question, and the first two parts usually scored full marks. The expression of D in part (iii) caused some problems with inaccurate algebra which then made the last two results unobtainable. Those that simplified D most neatly were in a stronger position to finish the question, though "if and only if" was frequently ignored, or only lip-service was paid to it. Consequently, scores were well-spread.