Year: 2009
Paper: 2
Question Number: 6
Course: LFM Pure and Mechanics
Section: Arithmetic and Geometric sequences
Of the 1000+ entries for this paper, around 920 scripts actually arrived for marking, giving another slight increase in the take-up for this paper. Of this number, five candidates scored a maximum and seventy-five achieved a scoring total of 100 or more. At the other end of the scale, almost two hundred candidates failed to reach the 40-mark mark. Otherwise, marks were spread reasonably normally across the mark range, though there were two peaks at about 45 and 65 in the distribution. It is comforting to find that the 'post-match analysis' bears out the view that I gained, quite firmly, during the marking process that there were several quantum states of mark-scoring ability amongst the candidature. Many (about one-fifth of the entry) struggled to find anything very much with which they were comfortable, and marks for these candidates were scored in 3s and 4s, with such folk often making eight or nine poor efforts at different questions without ever getting to grips with the content of any one of them. The next "ability band" saw those who either scored moderately well on a handful of questions or managed one really successful question plus a few bits-'n'-pieces in order to get up to a total in the mid-forties. To go much beyond that score required a little bit of extra talent that could lead them towards the next mark-hurdle in the mid-sixties. Thereafter, totals seemed to decline almost linearly on the distribution. Once again, it is clear that candidates need to give the questions at least a couple of minutes' worth of thought before commencing answering. Making attempts at more than the six scoring efforts permitted is a waste of valuable time, and the majority of those who do so are almost invariably the weaker brethren in the game. Many such candidates begin their efforts to individual questions promisingly, but get no more than half-a-dozen marks in before abandoning that question in favour of another – often with the replacement faring no better than its predecessor. In many such cases, the candidate's best-scoring question mark would come from their fifth, or sixth, or seventh, or …?, question attempted, and this suggests either that they do not know where their strengths lie, or that they are just not going to be of the view that they are not going to be challenged to think. And, to be fair to the setting panel, we did put some fairly obvious signposts up for those who might take the trouble to look for such things. With the pleasing number of very high totals to be found, it is clear that there are many places in which good marks were available to those with the ability to first identify them and then to persevere long enough to be able to determine what was really going on therein. It is extremely difficult to set papers in which each question is pitched at an equivalent level of difficulty. Apart from any other factors, candidates have widely differing strengths and weaknesses; one student's algebraic nuance can be the final nail in the coffin of many others, for instance. Moreover, it has seemed enormously clear to me – more particularly so since the arrival of modular A-levels – that there is absolutely no substitute for prolonged and determined practice at questions of substance. One moment's recognition of a technique at work can turn several hours of struggle into just a few seconds of polishing off, and a lack of experience is always painfully clear when marking work from candidates who are under-practised at either the art of prolonged mathematics or the science of creative problem-solving. At the other, more successful, end of the scale there were many candidates who managed to produce extraordinary amounts of outstanding work, racking up full-, or nearly full-, marks on question after question. With the marks distributed as they were, it seems that the paper was pitched appropriately at the intended level, and that it successfully managed to distinguish between the different ability-levels to be found among the candidates. 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. Moreover, many of these were clearly acts of desperation.
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1516.0
Banger Comparisons: 1
The Fibonacci sequence $F_1$, $F_2$, $F_3$, $\ldots$ is defined by $F_1=1$, $F_2= 1$ and
\[
F_{n+1} = F_n+F_{n-1} \qquad\qquad (n\ge 2).
\]
Write down the values of $F_3$, $F_4$, $\ldots$, $F_{10}$. Let $\displaystyle S=\sum_{i=1}^\infty \dfrac1 {F_i}\,$.
\begin{questionparts}
\item Show that $\displaystyle \frac 1{F_i} > \frac1{2F_{i-1}}\,$ for $i\ge4$ and deduce that $S > 3\,$. Show also that $S < 3\frac23\,$.
\item Show further that $3.2 < S < 3.5\,$.
\end{questionparts}\begin{array}{c|r}
n & F_n \\ \hline
1 & 1 \\
2 & 1 \\
3 & 2 \\
4 & 3 \\
5 & 5 \\
6 & 8 \\
7 & 13 \\
8 & 21 \\
9 & 34 \\
10 & 55
\end{array}
\begin{questionparts}
\item Claim: $\frac1{F_i} > \frac1{2F_{i-1}}$ for $i \geq 4$.
Proof: Since $F_i = F_{i-1}+F_{i-2}$ and $F_i > 1$ for $i \geq 1$ we have $F_i > F_{i-1}$ for $i \geq 3$. In particular we have $F_i = F_{i-1}+F_{i-2} < 2F_{i-1}$ for $i -1 \geq 3$ or $i \geq 4$.
Therefore $\frac{1}{F_i} > \frac1{2F_{i-1}}$
This was another very popular question, but the one with the lowest mean mark score of all the pure questions, at about 7. I think that the initial enthusiasm of seeing something familiar in the Fibonacci Numbers was more than countered by the inequalities work that formed the bulk of the question. Nonetheless, I suspect that, if given the opportunity to talk it through after the event, many candidates would admit that half of the marks on the question are actually ludicrously easy to acquire and that they were really only put-off by appearances. For instance, to show that S > any suitable lower-bound, one need only keep adding terms until the appropriate figure is exceeded. For those reciprocals of integers that are not easily calculated, such as 1/13, it is perfectly reasonable to note something that they are greater than and use that in its place. Thus, S ≥ 1 + 1 + 1/2 + 1/3 + 1/5 + 1/8 + ... ≥ 1 + 1 + 1/2 + 1/4 + 1/5 + 1/10 + ... = 2 + 0.5 + 0.25 + 0.2 + 0.1 = 3.05 > 3 works pretty easily (though it may not have scored full marks in (i) as a particular approach was requested), and something similar could be made to work in showing that S > 3.2 in (ii). The approach that the question was designed to direct candidates towards was that of stopping the direct calculation at some suitable stage, and using an inequality of the form 1/Fᵢ ≥ 1/Fᵢ², possibly alternating with the odd- and even-numbered terms, to make the remaining sum less than a summable infinite GP.