Introduction to Principle of Mathematical Induction

$1^2+2^2+3^2+...+n^2=?$,  $\forall n\in \mathrm{N}$.

Prove that $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\dots +\frac{1}{\sqrt{n}}>\sqrt{n}$ for all $n\ge 2$ and $n\in \mathrm{\mathbb{N}}$.

Prove that $2^n\le (n+1)!$ for all $n\in \mathrm{\mathbb{N}}$.

If a finite set has $n$ elements, prove by induction or otherwise that its power set has $2^n$ elements.

If $n>1$, prove that $n!<{\left(\frac{n+1}{2}\right)}^n$.

If $n$ straight lines are drawn on a plane such that no two are parallel and no three pass through the same point, then prove by mathematical induction that these $n$ straight lines divide the plane into $\frac{1}{2}\left(n^2+n+2\right)$ distinct regions.

Prove by induction that the number $2^{n^2}+1$ has $7$ in its units place for all integers $n\ge 2$.

Prove that, $n^2<n!$ for all integer $n\ge 4$.

Prove that $\frac{n^5}{5}+\frac{n^3}{3}+\frac{7n}{15}$ is always a positive integer whenever $n$ is so.

Use mathematical induction to show that ${25}^{n+1}-24n+5735$ is divisible by $(24{)}^2$ for all $n=1, 2, \dots$

Prove that ${11}^{n+2}+{12}^{2n+1}$ is divisible by $133$ for any integral non-negative $n$.

Prove that the inequality $\frac{1}{n+1}+\frac{1}{n+2}+\dots +\frac{1}{n+n}>\frac{13}{24}$ holds for any natural $n>1$.

Prove that the equality $\left(1-\frac{4}{1}\right)\left(1-\frac{4}{9}\right)\left(1-\frac{4}{25}\right)··\left(1-\frac{4}{(2n-1{)}^2}\right)=\frac{1+2n}{1-2n}$ holds true for any natural number $n$.

If $n$ is any $+ve$ integer or zero, show that $x^{3^n}+y^{3^n}$ is divisible by $x+y$.

Prove the following inequalities by induction on $n$ :

${10}^{n-2}>81n$ for all integers $n\ge 5$.

For what natural numbers $n$ is the inequality $2^n>n^2$ valid?

Prove the following inequalities by induction on $n$ :

$\frac{1}{n+1}+\frac{1}{n+2}+\dots +\frac{1}{3n+1}>1$ for all natural numbers $n$.

Prove the following inequalities by induction on $n$ :

$2^{n+1}<1+(n+1{)}^{2n}$ for all integers $n\ge 1$.

Prove the following inequalities by induction on $n$ :

$\frac{1}{1^2}+\frac{1}{2^2}+\dots +\frac{1}{n^2}\le 2-\frac{1}{n}$ for all positive integers $n$.

Prove the following inequalities by induction on $n$ :

$2^n>n$ for all integers $n\ge 0$.