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Magazine Problem 1
Prove the following by using the principle of mathematical induction for all \(n \in \mathbf{N}\). $$ 1+3+3^{2}+\ldots+3^{n-1}=\frac{\left(3^{n}-1\right)}{2} $$
Problem 2
Prove the following by using the principle of mathematical induction for all \(n \in \mathbf{N}\). $$ 1^{3}+2^{3}+3^{3}+\ldots+n^{3}=\left(\frac{n(n+1)}{2}\right)^{2} $$
Problem 4
Prove the following by using the principle of mathematical induction for all \(n \in \mathbf{N}\). $$ 1.2 .3+2.3 .4+\ldots+n(n+1)(n+2)=\frac{n(n+1)(n+2)(n+3)}{4} $$
Problem 5
Prove the following by using the principle of mathematical induction for all \(n \in \mathbf{N}\). $$ 1.3+2.3^{2}+3.3^{3}+\ldots+n \cdot 3^{n}=\frac{(2 n-1) 3^{n+1}+3}{4} $$
Problem 6
Prove the following by using the principle of mathematical induction for all \(n \in \mathbf{N}\). $$ 1.2+2.3+3.4+\ldots+n \cdot(n+1)=\left[\frac{n(n+1)(n+2)}{3}\right] $$
Problem 7
Prove the following by using the principle of mathematical induction for all \(n \in \mathbf{N}\). $$ 1.3+3.5+5.7+\ldots+(2 n-1)(2 n+1)=\frac{n\left(4 n^{2}+6 n-1\right)}{3} $$
Problem 8
Prove the following by using the principle of mathematical induction for all \(n \in \mathbf{N}\). $$ 1.2+2.2^{2}+3.2^{3}+\ldots+n .2^{n}=(n-1) 2^{n+1}+2 . $$
Problem 9
Prove the following by using the principle of mathematical induction for all \(n \in \mathbf{N}\). $$ \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots+\frac{1}{2^{n}}=1-\frac{1}{2^{n}} $$
Problem 10
Prove the following by using the principle of mathematical induction for all \(n \in \mathbf{N}\). $$ \frac{1}{2.5}+\frac{1}{5.8}+\frac{1}{8.11}+\ldots+\frac{1}{(3 n-1)(3 n+2)}=\frac{n}{(6 n+4)} . $$
Problem 11
Prove the following by using the principle of mathematical induction for all \(n \in \mathbf{N}\). $$ \frac{1}{1.2 .3}+\frac{1}{2.3 .4}+\frac{1}{3.4 .5}+\ldots+\frac{1}{n(n+1)(n+2)}=\frac{n(n+3)}{4(n+1)(n+2)} $$
