# Recent questions tagged exercise-4.1

### 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)$$=(n+1)^2 ### Prove the following by using the principle of mathematical induction for all n \in N  a+ar+ar^2+...+ar^{n-1}=\large\frac{a(r^n-1)}{r-1} ### Prove the following by using the principle of mathematical induction for all n \in N  \large\frac{1}{1.2.3}$$+\large\frac{1}{2.3.4}$$+\large\frac{1}{3.4.5}$$+...+\large\frac{1}{n(n+1)(n+2)}$$=\large\frac{n(n+3)}{4(n+1)(n+2)} ### Prove the following by using the principle of mathematical induction for all n \in N  \large\frac{1}{2.5}$$+\large\frac{1}{5.8}$$+\large\frac{1}{8.11}$$+...+\large\frac{1}{(3n-1)(3n+2)}$$=\large\frac{1}{(6n+4)} ### Prove the following by using the principle of mathematical induction for all n \in N  \large\frac{1}{2}$$+ \large\frac{1}{4}$$+\large\frac{1}{8}$$+...+\large\frac{1}{2^n}$$=1-\large\frac{1}{2^n} ### Prove the following by using the principle of mathematical induction for all n \in N  1.2+2.2^2+3.2^2+...+n.2^n=(n-1)2^{n+1}+2 ### Prove the following by using the principle of mathematical induction for all n \in N  1.3+3.5+5.7+...+(2n-1)(2n+1)= \large\frac{n(4n^2+6n-1)}{3} ### Prove the following by using the principle of mathematical induction for all n \in N  1.2+2.3+3.4+...+n.(n+1) \bigg[ \large\frac{n(n+1)(n+2)}{3} \bigg] ### Prove the following by using the principle of mathematical induction for all n \in N  1.3+2.3^2+3.3^3+...+n.3^n=\large\frac{(2n-1)3^{n+1}+3}{4} ### Prove the following by using the principle of mathematical induction for all n \in N  1.2.3 + 2.3.4+...+n(n+1)(n+2)= \large\frac{n(n+1)(n+2)(n+3)}{4} ### Prove the following by using the principle of mathematical induction for all n \in N  1+ \large\frac{1}{(1+2)}$$+ \large\frac{1}{(1+2+3)}$$+...+ \large\frac{1}{(1+2+3...n)}$$= \large\frac{2n}{(n+1)}$

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