G Tewani Solutions for Chapter: Principle of Mathematical Induction, Exercise 1: Exercises

Prove using the principle of mathematical induction that for all $n\in N$, $\frac{1}{1·2·3}+\frac{1}{2·3·4}+\frac{1}{3·4·5}+\dots +\frac{1}{n(n+1)(n+2)}=\frac{n(n+3)}{4(n+1)(n+2)}$

Using principle of mathematical induction, Prove that $\forall n\in \mathrm{\mathbb{N}}, 2^{3n}-1$ is divisible by $7$.

Using principle of mathematical induction, prove that for all $n\in N, x^n-y^n$ is divisible by $x-y,$ where $x$ and $y$ are any integers such that $x\ne y.$

Using principle of mathematical induction, prove that for all $n\in N, n\ge 2, \sqrt{n}< \frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{n}}.$

Using principle of mathematical induction, prove that for all $n\in N, \frac{n^5}{5}+\frac{n^3}{3}+\frac{7n}{15}$ is a natural number.

Using mathematical induction, prove that $7^{4^{{}^n}}-1$ is divisible by $2^{2n+3}$ for any natural number $n$

Prove by mathematical induction that $n^5$ and $n$ have the same unit digit for any natural number $n.$

A sequence $b_0,b_1,b_2,....$ is defined by letting $b_0=5$ and $b_k=4+b_{k-1},$ for all natural numbers $k$. Show that $b_n=5+4n,$ for all natural numbers $n$ using mathematical induction.