Year: 2015
Paper: 3
Question Number: 2
Course: LFM Pure
Section: Proof
A very similar number of candidates to 2014 once again ensured that all questions received a decent number of attempts, with seven questions being very popular rather than five being so in 2014, but the most popular questions were attempted by percentages in the 80s rather than 90s. All but one question was answered perfectly at least once, the one exception receiving a number of very close to perfect solutions. About 70% attempted at least six questions, and in those cases where more than six were attempted, the extra attempts were usually fairly superficial.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1529.7
Banger Comparisons: 4
If $s_1$, $s_2$, $s_3$, $\ldots$ and $t_1$, $t_2$, $t_3$, $\ldots$ are sequences of positive numbers, we write
\[
(s_n)\le (t_n)
\]
to mean
\begin{center}
"there exists a positive integer $m$ such that $s_n \le t_n$ whenever $n\ge m$".
\end{center}
Determine whether each of the following statements is true or false. In the case of a true statement, you should give a proof which includes an explicit determination of an appropriate $m$; in the case of a false statement, you should give a counterexample.
\begin{questionparts}
\item $(1000n) \le (n^2)\,$.
\item If it is not the case that $(s_n)\le (t_n)$, then it is the case that $(t_n)\le (s_n)\,$.
\item If $(s_n)\le (t_n)$ and $(t_n) \le (u_n)$, then $(s_n)\le (u_n)\,$.
\item $(n^2)\le (2^n)\,$.
\end{questionparts}\begin{questionparts}
\item If $m = 1000$, then $n \geq m \Rightarrow n^2 \geq 1000n \Rightarrow (1000n) \leq (n^2)$
\item This is false. Let $s_i = 1,2,1,2,\cdots$ and $t_i = 2,1,2,1,\cdots$.
\item Suppose that for $n \geq m_1, s_n \le t_n$ and for $n \geq m_2, s_t \le u_n$, then for $n \geq m = \max(m_1, m_2), s_n \leq t_n \leq u_n \Rightarrow s_n \leq u_n \Rightarrow (s_n) \leq (u_n)$
\item Let $m = 6$, then if $n \geq m, 2^n \geq 1 + n + \frac{n(n-1)}{2} + \frac{n(n-1)}{2} + n + 1 = n^2 + n + 2 \geq n^2$, so $(2^n) \geq (n^2)$
\end{questionparts}
Nearly three quarters attempted this, though again with moderate success as the main feature of the question was proof, and this was frequently handled cavalierly. Whilst it was not a crucial aspect of the question, ignoring the fact that the question deals with sequences of positive numbers was careless. Answers to the first part suffered at times from lack of argument or backwards logic. Part (ii) was generally well answered, although there were some silly counter-examples. This part suffered from those who completely missed the point of what the question was all about, forgetting the initial definition. Whilst most appreciated that part (iv) was true, there were many different methods used to attempt to prove it, and often unsuccessfully. Whilst induction using algebra is fairly straightforward, differentiation with or without logarithms and graphical methods frequently came to grief.