The Principle of Mathematical Induction

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

Define well ordering principle in mathematical induction.

Prove the following by principle of mathematical induction $\forall$ $n\in N$

$1+\frac{1}{1+2}+\frac{1}{1+2+3}+\dots +\frac{1}{1+2+3+\dots +n}=\frac{2n}{n+1}$

Define well ordering principle in mathematical induction.

Prove the following by principle of mathematical induction $\forall n\in N$ 

$\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+\dots ..+\frac{1}{n(n+1)(n+2)}=\frac{n(n+3)}{4(n+1)(n+2)}$

Define well ordering principle in mathematical induction.

Prove the following by the principle of mathematical induction for all $n\in N:$

$1.2+2.2^2+3.2^3+\dots \dots +n.2^n=(n-1)2^{n+1}+2$

Define well ordering principle in mathematical induction.

Prove the following by the principle of mathematical induction for all $n\in N:$

$1.3+2.3^2+3.3^3+\dots ..+n.3^n=\frac{(2n-1)3^{n+1}+3}{4}$

Define well ordering principle in mathematical induction.

Prove that $(1+x{)}^n\ge 1+nx,$ for all natural numbers $n,$ where $x>-1$

If $\mathrm{P}\left(n\right):2^n<n!, n\in \mathrm{N},$ then $\mathrm{P}\left(n\right)$ is true for:

If $x^n-1$ is divisible by $x-k,$ then the least positive integral value of $k$ is

Let $P\left(n\right)$ be a statement and let $P\left(n\right)\to P\left(n+1\right)$ is true for all natural numbers $n$, then $P\left(n\right)$ is true

$x^n+y^n$ is divisible by

If $1.2+2.3+3.4+\dots .+n(n+1)=\frac{n(n+1)(n+2)}{3}=P\left(n\right)$, then 

If  $3.6+6.9+9.12+\dots ..+3n(3n+3)$$=3n(n+1)(n+2)=P\left(n\right)$ then 

For all $n\in N,$ $n(n+1)(n+5)$ is a multiple of 

Inequality $3^n<(n+1)!, n\in N$

For all $n\in N$,  $2.5+5.8+8.11+\dots .+(3n-1)(3n+2)=n\left(3n^2+6n+1\right)=P\left(n\right)$ then 

If we take any three consecutive natural numbers, then the sum of their cubes is always divisible by $$

For every natural number $n$,  $n^3+(n+1{)}^3+(n+2{)}^3$ is divisible by  

For all $n\in N,$ $1·2+2·2^2+3·2^3+\dots \dots +n·2^n=$

For all $n\in N,$ $1.3+2.3^2+3.3^3+\dots ..+n.3^n=$

The statement $:2^n>3n$, $n\in N$.