A P Prabhakaran Solutions for Chapter: Principle of Mathematical Induction, Exercise 2: SELF EVALUATION TEST

Prove by the principle of mathematical induction that all $n\in N,$ $1^3+2^3+3^3+\cdots n^3={\left[\frac{n(n+1)}{2}\right]}^2$

If $P\left(n\right)$ is the statement $n(n+1)(n+2)$ is divisible by $12$, show that $P\left(3\right)$ and $P( 4)$ are true but not $P\left(5\right)$.

 Using the principle of mathematical induction, prove that ${10}^{2n-1}+1$ is divisible by $11$ for all $n\in N$.
 

.Use mathematical induction to prove that 

$\left(1+\frac{3}{1}\right)·\left(1+\frac{5}{4}\right)\cdots \left(1+\frac{2n+1}{n^2}\right)=(n+1{)}^2$,for all $n\in N$.

Using principle of mathematical induction, prove that $a+\left(a+d\right)+\left(a+2 d\right)+\dots +\left[a+\left(n-1\right) d\right]=\frac{n}{2}\left[2 a+\left(n-1\right)d\right]$

Prove by the mathematical induction that the sum of the cubes of three consecutive natural number is divisible by $9.$

Prove by mathematical induction that $\left(x^n-y^n\right)$ is divisible by $\left(x-y\right)$, for all $x\in N$

Prove using the principle of mathematical induction for all $n\in \mathrm{N}$ that

$1·3+3·5+5·7+\cdots +(2n-1)(2n+1)=\frac{n\left(4n^2+6n-1\right)}{3}$