Using the principle of mathematical induction, prove the following for $n\in N$;

$1+\frac{1}{\left(1+2\right)}+\frac{1}{\left(1+2+3\right)}+\dots +\frac{1}{\left(1+2+3+\dots +n\right)}=\frac{2n}{\left(n+1\right)}$

Using the principle of mathematical induction, prove the following for $n\in N$;

$\frac{1}{2·5}+\frac{1}{5·8}+\frac{1}{8·11}+\dots +\frac{1}{\left(3n-1\right)\left(3n+2\right)}=\frac{n}{\left(6n+4\right)}$

Using the principle of mathematical induction, prove the following for $n\in N$;

$\frac{1}{1·4}+\frac{1}{4·7}+\frac{1}{7·10}+\dots +\frac{1}{\left(3n-2\right)\left(3n+1\right)}=\frac{n}{\left(3n+1\right)}$

Using the principle of mathematical induction, prove each of the following for $n\in N$;

$\frac{1}{1·3}+\frac{1}{3·5}+\frac{1}{5·7}+...+\frac{1}{\left(2n-1\right)\left(2n+1\right)}=\frac{n}{\left(2n+1\right)}$

Using the principle of mathematical induction, prove the following for $n\in N$;

$\frac{1}{2·5}+\frac{1}{5·8}+\frac{1}{8·11}+\dots +\frac{1}{\left(3n-1\right)\left(3n+2\right)}=\frac{n}{6n+4}$

Using the principle of mathematical induction, prove the following for undefined $n\in N$

$\left(1+\frac{3}{1}\right)\left(1+\frac{5}{4}\right)\left(1+\frac{7}{9}\right)\dots \left\{1+\frac{\left(2n+1\right)}{n^2}\right\}={\left(n+1\right)}^2$

Using the principle of mathematical induction, prove the following for $n\in N$;

$\left(1+\frac{1}{1}\right)\left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right)\dots \left(1+\frac{1}{n}\right)=\left(n+1\right)$

Using the principle of mathematical induction, prove the following for $n\in N$;

$n\left(n+1\right)\left(n+2\right)$ is a multiple of $6$.

Using the principle of mathematical induction, prove the following for $n\in N$;

$\left(x^{2n}-y^{2n}\right)$ is divisible by $\left(x+y\right)$.