# Proof By Induction Inequalities Questions, Answers and Solutions

a - answer    s - solution    v - video

## Question 1

Prove by mathematical induction:

a) $2^n>2n$ for all $n>2$ and $n\in\mathbb{Z}^+$a s v

b) $2^n<3^n$ for $n\geq1$ and $n\in\mathbb{Z}^+$a s v

c) $n!>2^n$ for $n\geq4$a s v

d) $2^n>4n$ for $n\geq5$ and $n\in\mathbb{Z}^+$a s v

e) $n^2\ge2n$ for $n>1$ and $n\in\mathbb{Z}^+$a s v

f) $n^2<4^{n-1}$ for $n\geq3$ and $n\in\mathbb{Z}^+$a s v

## Question 2

Prove by mathematical induction:

a) $1+\frac{1}{\sqrt2}+\frac{1}{\sqrt3}\dotsb+\frac{1}{\sqrt n}<2\sqrt n-1$ for $n\ge2$ and $n\in\mathbb{Z}^+$a s v