Section 1
Introduction
Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.$$1+3+3^{2}+\ldots+3^{n-1}=\frac{\left(3^{n}-1\right)}{2}$$
Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.$$1^{3}+2^{3}+3^{3}+\ldots+n^{3}=\left(\frac{n(n+1)}{2}\right)^{2}$$
Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.$$1+\frac{1}{(1+2)}+\frac{1}{(1+2+3)}+\ldots+\frac{1}{(1+2+3+\ldots n)}=\frac{2 n}{(n+1)}$$
Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.$$1.2 .3+2.3 .4+\ldots+n(n+1)(n+2)=\frac{n(n+1)(n+2)(n+3)}{4}$$
Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.$$1.3+2.3^{2}+3.3^{3}+\ldots+n \cdot 3^{n}=\frac{(2 n-1) 3^{n+1}+3}{4}$$
Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.$$1.2+2.3+3.4+\ldots+n \cdot(n+1)=\left[\frac{n(n+1)(n+2)}{3}\right]$$
Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.$$1.3+3.5+5.7+\ldots+(2 n-1)(2 n+1)=\frac{n\left(4 n^{2}+6 n-1\right)}{3}$$
Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.$$1.2+2.2^{2}+3.2^{3}+\ldots+n .2^{n}=(n-1) 2^{n+1}+2 .$$
Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.$$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots+\frac{1}{2^{n}}=1-\frac{1}{2^{n}}$$
Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.$$\frac{1}{2.5}+\frac{1}{5.8}+\frac{1}{8.11}+\ldots+\frac{1}{(3 n-1)(3 n+2)}=\frac{n}{(6 n+4)} .$$
Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.$$\frac{1}{1.2 .3}+\frac{1}{2.3 .4}+\frac{1}{3.4 .5}+\ldots+\frac{1}{n(n+1)(n+2)}=\frac{n(n+3)}{4(n+1)(n+2)}$$
Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.$$a+a r+a r^{2}+\ldots+a r^{n-1}=\frac{a\left(r^{n}-1\right)}{r-1} .$$
Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.$$\left(1+\frac{3}{1}\right)\left(1+\frac{5}{4}\right)\left(1+\frac{7}{9}\right) \ldots\left(1+\frac{(2 n+1)}{n^{2}}\right)=(n+1)^{2} .$$
Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.$$\left(1+\frac{1}{1}\right)\left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right) \ldots\left(1+\frac{1}{n}\right)=(n+1)$$
Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.$$1^{2}+3^{2}+5^{2}+\ldots+(2 n-1)^{2}=\frac{n(2 n-1)(2 n+1)}{3}$$
Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.$$\frac{1}{1.4}+\frac{1}{4.7}+\frac{1}{7.10}+\ldots+\frac{1}{(3 n-2)(3 n+1)}=\frac{n}{(3 n+1)}$$
Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.$$\frac{1}{3.5}+\frac{1}{5.7}+\frac{1}{7.9}+\ldots+\frac{1}{(2 n+1)(2 n+3)}=\frac{n}{3(2 n+3)}$$
Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.$$1+2+3+\ldots+n<\frac{1}{8}(2 n+1)^{2} .$$
Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.$$n(n+1)(n+5) \text { is a multiple of } 3 .$$
Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.$$10^{2 n-1}+1 \text { is divisible by } 11 .$$
Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.$$x^{2 n}-y^{2 n} \text { is divisible by } x+y \text { . }$$
Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.$$3^{2 n+2}-8 n-9 \text { is divisible by } 8 \text { . }$$
Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.$$41^{n}-14^{n} \text { is a multiple of } 27 \text { . }$$
Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.$$(2 n+7)<(n+3)^{2}$$