Year: 2020
Paper: 3
Question Number: 8
Course: UFM Additional Further Pure
Section: Sequences and Series
No solution available for this problem.
In spite of the change to criteria for entering the paper, there was still a very healthy number of candidates, and the vast majority handled the protocols for the online testing very well. Just over half the candidates attempted exactly six questions, and whilst about 10% attempted a seventh, hardly any did more than seven. With 20% attempting five questions, and 10% attempting only four, overall, there were very few candidates not attempting the target number. There was a spread of popularity across the questions, with no question attracting more than 90% of candidates and only one less than 10%, but every question received a good number of attempts. Likewise, there was a spread of success on the questions, though every question attracted at least one perfect solution.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
A sequence $u_k$, for integer $k \geqslant 1$, is defined as follows.
\[ u_1 = 1 \]
\[ u_{2k} = u_k \text{ for } k \geqslant 1 \]
\[ u_{2k+1} = u_k + u_{k+1} \text{ for } k \geqslant 1 \]
\begin{questionparts}
\item Show that, for every pair of consecutive terms of this sequence, except the first pair, the term with odd subscript is larger than the term with even subscript.
\item Suppose that two consecutive terms in this sequence have a common factor greater than one. Show that there are then two consecutive terms earlier in the sequence which have the same common factor. Deduce that any two consecutive terms in this sequence are co-prime (do not have a common factor greater than one).
\item Prove that it is not possible for two positive integers to appear consecutively in the same order in two different places in the sequence.
\item Suppose that $a$ and $b$ are two co-prime positive integers which do not occur consecutively in the sequence with $b$ following $a$. If $a > b$, show that $a-b$ and $b$ are two co-prime positive integers which do not occur consecutively in the sequence with $b$ following $a-b$, and whose sum is smaller than $a+b$. Find a similar result for $a < b$.
\item For each integer $n \geqslant 1$, define the function $\mathrm{f}$ from the positive integers to the positive rational numbers by $\mathrm{f}(n) = \dfrac{u_n}{u_{n+1}}$. Show that the range of $\mathrm{f}$ is all the positive rational numbers, and that $\mathrm{f}$ has an inverse.
\end{questionparts}
About 60% of the candidates attempted this question, but it was the second least successful question on the paper with a mean score of about one third marks. There were some very good solutions to this question, but most candidates only provided fragmentary answers, and stopped after the first couple of parts. A common mistake was to use the condition u_{2k} = u_k along with u_1 = 1 to erroneously conclude that all the terms with an even subscript are equal to 1. This might have been avoided if the candidates had written out the first 10 (or so) terms of the sequence to help them get a "feel" for what was happening, which could have also help stopped some other misconceptions along the way. Part (i) was generally well done, but some candidates did not consider both cases u_{2k+1} > u_{2k} and u_{2k−1} > u_{2k}. Other candidates concluded that u_{k+1} + u_k > u_k without justifying this inequality by stating that all the terms are positive. Part (ii) was attempted well by many candidates but was less successful than part (i). Some candidates who correctly considered both the (u_{2k−1}, u_{2k}) and (u_{2k}, u_{2k+1}) cases in part (i) then failed to consider both in this part. Some candidates erroneously assumed that if two terms p, q share a common factor and p < q then it must be the case that q = kp. Part (iii) was only answered well by a few of the candidates. Some did not appreciate that "consecutively" means appears one directly after another, instead taking it to mean that the second one occurs at some position after the first one. Only a small minority of attempts considered both the (u_{2k−1}, u_{2k}) and (u_{2k}, u_{2k+1}) cases. A lot of candidates erroneously stated that "if u_k = a then if a is going to reappear then the next index must be 2^n k for some integer n". A look at the first few terms of the sequence shows that u_5 = u_7 = 3 which contradicts that statement. Part (iv) was not well attempted. Some candidates did not process the wording (which was designed to help with the next part), and some tried to show instead that if a and b were two co-prime integers which do occur consecutively in the sequence etc. The most successful candidates used contradiction here to show that if a − b and b do occur consecutively then this means that a and b must occur consecutively. Some candidates correctly showed the first result, but when trying to find the similar result for a < b ended up with a following b and so essentially proved the same result again. Part (v) was answered by only a few of the candidates attempting this question. There were some very well-reasoned arguments, including some candidates who used a construction method to justify that all possible rational numbers are in the range of f(n). Only a very small number connected part (iv) to this part of the question.