Year: 2011
Paper: 2
Question Number: 7
Course: LFM Pure and Mechanics
Section: Arithmetic and Geometric sequences
No solution available for this problem.
There were just under 1000 entries for paper II this year, almost exactly the same number as last year. After the relatively easy time candidates experienced on last year's paper, this year's questions had been toughened up significantly, with particular attention made to ensure that candidates had to be prepared to invest more thought at the start of each question – last year saw far too many attempts from the weaker brethren at little more than the first part of up to ten questions, when the idea is that they should devote 25-40 minutes on four to six complete questions in order to present work of a substantial nature. It was also the intention to toughen up the final "quarter" of questions, so that a complete, or nearly-complete, conclusion to any question represented a significant (and, hopefully, satisfying) mathematical achievement. Although such matters are always best assessed with the benefit of hindsight, our efforts in these areas seem to have proved entirely successful, with the vast majority of candidates concentrating their efforts on four to six questions, as planned. Moreover, marks really did have to be earned: only around 20 candidates managed to gain or exceed a score of 100, and only a third of the entry managed to hit the half-way mark of 60. As in previous years, the pure maths questions provided the bulk of candidates' work, with relatively few efforts to be found at the applied ones. Questions 1 and 2 were attempted by almost all candidates; 3 and 4 by around three-quarters of them; 6, 7 and 9 by around half; the remaining questions were less popular, and some received almost no "hits". Overall, the highest scoring questions (averaging over half-marks) were 1, 2 and 9, along with 13 (very few attempts, but those who braved it scored very well). This at least is indicative that candidates are being careful in exercising some degree of thought when choosing (at least the first four) 'good' questions for themselves, although finding six successful questions then turned out to be a key discriminating factor of candidates' abilities from the examining team's perspective. Each of questions 4-8, 11 & 12 were rather poorly scored on, with average scores of only 5.5 to 6.6.
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
The two sequences $a_0$, $a_1$, $a_2$, $\ldots$
and
$b_0$, $b_1$, $b_2$, $\ldots$
have general terms
\[
a_n = \lambda^n +\mu^n
\text { \ \ \ and \ \ \ }
b_n = \lambda^n - \mu^n\,,
\]
respectively, where $\lambda = 1+\sqrt2$ and $\mu= 1-\sqrt2\,$.
\begin{questionparts}
\item Show that $\displaystyle \sum_{r=0}^nb_r = -\sqrt2 + \frac
1 {\sqrt2} \,a_{\low n+1}\,$,
and give a corresponding result for
$\displaystyle \sum_{r=0}^na_r\,$.
\item Show that, if $n$ is odd,
$$\sum_{m=0}^{2n}\left( \sum_{r=0}^m a_{\low r}\right)
= \tfrac12 b_{n+1}^2\,,$$
and give a corresponding result when $n$
is even.
\item Show that, if $n$ is even,
$$\left(\sum_{r=0}^na_r\right)^{\!2}
-\sum_{r=0}^n a_{\low 2r+1} =2\,,$$
and give a corresponding result when
$n$ is odd.
\end{questionparts}
The initial hurdle in this question involved little more than splitting the series into separate sums of powers of α and β, leading to easy sums of GPs. Many missed this and spent a lot of wasted time playing around algebraically without getting anywhere useful. In (ii), many candidates applied (i) once, for the inner summation, but then failed to do so again for the second time, and this was rather puzzling. Equally puzzling was the lack of recognition, amongst those who had completed most of the first two parts of the question successfully, that the sum of the odd terms in (iii) was still a geometric series. Almost exactly half of all candidates made an attempt at this question, but the average score was only just over 5/20.