Q. Prove the following by using the principle of mathematical induction for all $n\in N:1.2+2.3+3.4+......+n(n+1)=\left[\frac{n(n+1)(n+2)}{3}\right]$

Q. Prove the following by using the principle of mathematical induction for all $n\in N:1.3+2.3^2+3.3^3+.....+n.3{}^n=\frac{(2n-1)3^{n+1}+3}{4}$

Q. Prove the following by using the principle of mathematical induction for all $n\in N:P\left(n\right):a+ar+a{r}^2+......+a{r}^{n-1}=\frac{a(r^n-1)}{r-1}$

Q. Prove the following by using the principle of mathematical induction for all $n\in N:\frac{1}{1.4}+\frac{1}{4.7}+\frac{1}{7.10}+.....+\frac{1}{(3n-2)(3n+1)}=\frac{n}{(3n+1)}$

Q. Prove the following b y using the principle of mathematical induction for all $n\in N:1+2+3+.....+n<\frac{1}{8}{(2n+1)}^2$

Q. Prove the following by using the principle of mathematical induction for all $n\in N$
$1^3+2^3+3^3+.......+n^3={\left[\frac{n(n+1)}{2}\right]}^2$

Q. Prove the following by using the principle of mathematical induction for all $n\in N:\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+.....+\frac{1}{2^n}=1-\frac{1}{2^n}$

Q. Prove the following by using the principle of mathematical induction for all $n\in N:n(n+1)(n+5)$ is multiple of $3$.

Q. Prove the following by using the principle of mathematical induction for all $n\in N:\mathrm{1.2.3}+\mathrm{2.3.4}+......+n(n+1)(n+2)=\frac{n(n+1)(n+2)(n+3)}{4}$

Q. Prove the following by using principle of mathematical induction for all $n\in N:1.3+3.5+5.7+.......+(2n-1)(2n+1)=\frac{n(4n^2+6n-1)}{3}$

Q. Prove the following using the principle of mathematical induction for all $n\in N$:
$1+3+3^2+........+3^{n-1}=\frac{(3^n-1)}{2}$

Q. Prove the following by using the principle of mathematical induction for all $n\in$:
$1+\frac{1}{(1+2)}+\frac{1}{(1+2+3)}+..........+\frac{1}{1+2+3+.....n)}=\frac{2n}{(n+1)}$

Q. Prove the following by using the principle of mathematical induction for all $n\in N:(1+\frac{1}{1})(1+\frac{1}{2})(1+\frac{1}{3})......(1+\frac{1}{n})=(n+1)$

Q. Prove the following by using the principle of mathematical induction for all $n\in N:(1+\frac{3}{1})(1+\frac{5}{4})(1+\frac{7}{9})......(1+\frac{(2n+1)}{n^2})={(n+1)}^2$

Q. Prove the following by using the principle of mathematical induction for all $n\in N:{10}^{2n-1}+1$ is divisible by $11$

Q. Prove the following by using the principle of mathematical induction for all $n\in N:\frac{1}{\mathrm{1.2.3}}+\frac{1}{\mathrm{2.3.4}}+\frac{1}{\mathrm{3.4.5}}+.....+\frac{1}{n(n+1)(n+2)}=\frac{n(n+3)}{4(n+1)(n+2)}$

Q. Prove the following by using the principle of mathematical induction for all $n\in N:1.2+2.2^2+3.2^2+.......+n.2^n=(n-1)2^{n+1}+2$

Q. Prove the following by using the principle of mathematical induction for all $n\in N:\frac{1}{2.5}+\frac{1}{5.8}+\frac{1}{8.11}+......+\frac{1}{(3n-1)(3n+2)}=\frac{n}{(6n+4)}$

Q. Prove the following by using the principle of mathematical induction for all $n\in N:1^2+3^2+5^2+.......+({2n-1)}^2=\frac{n(2n-1)(2n+1)}{3}$

Q. Prove the following by using principle of mathematical induction for all $n\in N:\frac{1}{3.5}+\frac{1}{5.7}+\frac{1}{7.9}+.....+\frac{1}{(2n+1)(2n+3)}=\frac{n}{3(2n+3)}$