Further Mathematics 9231: Proof by induction Topical Past Paper Questions

The sequence of positive numbers $ u_1, u_2, u_3, \ldots $ is such that $ u_1 > 1 $ and, for $ n \geq 1 $, $$ u_{n+1}=\frac{u_{n}^{2}+2u_{n}+2}{u_{n}+4}. $$ (a) By considering $u_{n+1}-1$, or otherwise, prove by mathematical induction that $u_n>1$ for all positive integers $n$. (…

Prove by mathematical induction that $ \left(\frac{6}{5}\right)^{n} \geqslant 1 + \frac{1}{5} n $ for all positive integers $ n $.

Prove by mathematical induction that, for all positive integers n, $ \frac{\mathrm{d}^{n}}{\mathrm{d}x^{n}}\left(\tan^{-1}x\right)=P_{n}(x)\left(1+x^{2}\right)^{-n} $, where P_{n}(x) is a polynomial of degree n-1. [6]