# Recent questions and answers in Principle of Mathematical Induction

Questions from: Principle of Mathematical Induction

### Prove the following by using the principle of mathematical induction for all n ∈ N:  $1+2+3+...+n<\large\frac{1}{8}$$(2n+1)^2 ### Prove the following by using the principle of mathematical induction for all n \in N  \large\frac{1}{3.5}$$+\large\frac{1}{5.7}$$+\large\frac{1}{7.9}$$+...+\large\frac{1}{(2n+1)(2n+3)}$$=\large\frac{n}{3(2n+3)} ### Prove the following by using the principle of mathematical induction for all n \in N  \large\frac{1}{1.4}$$+\large\frac{1}{4.7}$$+\large\frac{1}{7.10}$$+...+\large\frac{1}{(3n-2)(3n+1)}$$=\large\frac{n}{(3n+1)} ### Prove the following by using the principle of mathematical induction for all n \in N  1^2+3^2+5^2+...+(2n-1)^2=\large\frac{n(2n-1)(2n+1)}{3} ### Prove the following by using the principle of mathematical induction for all n \in N  \bigg( 1+\large\frac{1}{1} \bigg)$$\bigg( 1+\large\frac{1}{2} \bigg)$$\bigg( 1+\large\frac{1}{3} \bigg)$$...\bigg( 1+\large\frac{1}{n} \bigg)$$=(n+1) ### Prove the following by using the principle of mathematical induction for all n \in N  \bigg( 1+ \large\frac{3}{1} \bigg)$$\bigg( 1+ \large\frac{5}{4} \bigg)$$\bigg( 1+ \large\frac{7}{9} \bigg)$$...\bigg( 1+ \large\frac{(2n+1)}{n^2} \bigg)$$=(n+1)^2$

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