This is an optional module for those preparing for Engineering examinations
Q-1) Do there exist distinct positive integers \(a,b,c\) such that \(a,b,c,b+c-a,c+a-b,a+b-c,a+b+c\) form an arithmetic progression (in some order)?
Q-2) Define a sequence \(\left\{a_n\right\}_{n \ge 0}\) by \(a_0=0\), \(a_1=1\) and
\(a_n=2a_{n-1}+a_{n-2}\) for \(n \ge 2\).
a) Show that for \(m>0\) and \(0\le j \le m\), \(2a_m\) divides \(a_{m+j}+(-1)^ja_{m-j}\)
b) If \(2^k\) divides \(n\), show that \(2^k\) divides \(a_n\).
Q-3) Consider a nonconstant arithmetic progression \(a_1,a_2,…\). Suppose there exist relatively prime positive integers \(p>1\) and \(q>1\) such that \(a_1^2,a_{p+1}^2,a_{q+1}^2\) are also terms of the same A.P. Show that the terms of the A.P are all positive integers.
Q-4) Given a real number \(a\), we define a sequence by \(x_0=1, x_1=x_2=a\) and \(x_{n+1}=2x_nx_{n-1}-x_{n-2}\) for \(n\ge 2\). Show that if \(x_n=0\) for some \(n\), then the sequence is periodic.
Q-5) Suppose that \(f(x)=\sum\limits_{i=0}^\infty c_ix^i\) is a power series for which each coefficient \(c_i\) is \(0\) or \(1\). Show that if \(f\left(\frac{2}{3}\right)=\frac{3}{2}\), then \(f\left(\frac{1}{2}\right)\) must be irrational.
Q-6) Let \(a_0=\frac{5}{2}\) and \(a_k=a_{k-1}^2-2\) for \(k \ge 1\).
a) Show that \(a_k=\frac{2^{2^{k+1}}+1}{2^{2^k}}\)
b) Using a) or otherwise find the value of \(\prod\limits_{k=0}^\infty\left(1-\frac{1}{a_k}\right)\)
Q-7) We define a sequence by \(a_1=1, a_{2n}=a_n\) and \(a_{2n+1}=(-1)^na_n\).
Find the value of \(\sum\limits_{n=1}^{2013}a_na_{n+2}\).
Q-8) Is there an infinite sequence of real numbers \(\left\{a_n\right\}\) such that
\(a_1^m+a_2^m+a_3^m+….=m\) for every positive integer \(m\)?
Q-9) Let \(1,2,3,..2005,2006,2007,2009,2012,2016\) be a sequence defined by \(x_k=k\) for \(k=1,2,..,2006\) and \(x_{k+1}=x_k+x_{k-2005}\) for \(k\ge 2006\). Show that the sequence has \(2005\) consecutive terms each divisible by \(2006\).
Q-10) Let \(n\) be an odd positive integer and let \(\theta\) be a real number such that \(\frac{\theta}{\pi}\) is rational. Set
\(a_k=\tan (\theta+\frac{k\pi}{n})\) for all \(k \in \mathbb{N}\)
Show that \(\frac{a_1+a_2+…a_n}{a_1a_2…a_n}\) is an integer and determine its value.
(We have not yet covered trigonometric sums, so don’t panic if you’re unable to solve this question)