Prove that the sum of first n even numbers is n(n+1).

Prove that 2+4+6+8+.....2=n(n+1).

Using principle of mathematical induction, prove that \(\left( 1+\frac { 1 }{ 1 } \right) \left( 1+\frac { 1 }{ 2 } \right) \left( 1+\frac { 1 }{ 3 } \right) .....\left( 1+\frac { 1 }{ n } \right) =n+1\)

Prove that \(\sum _{ t=1 }^{ n-1 }{ t(t+1) } =\frac { n(n-1)(n+1) }{ 3 } \) , for all natural numbers \(n\ge 2\)

Prove that 2ⁿ>n for all positive integers n.

Prove by the principle of mathematical induction that \({ 3 }^{ n }>{ 2 }^{ n }\) , for all \(n\in N\) .

Prove that \(1+2+3......n<\frac { 1 }{ 8 } { (2n+1) }^{ 2 }\) , for all natural numbers n.

Prove that \({(1+x) }^{ n }\ge (1+nx)\) , for all natural number n, where x > -1.

Prove by the principle of mathematical induction that \(1\times 1!+2\times 2!+3\times 3!+....+n\times n!=(n+1)!-1\)for all natural numbers n.

Let P(n) be the statement "3ⁿ">n". If p(n) is true, prove that p(n+1) is true.

Use the principle of mathematical induction to prove that \({ n }^{ 3 }-7n+3\) is divisible by 3, for all natural numbers of n.

Prove that \({ 1 }^{ 2 }{ +2 }^{ 2 }+....+{ n }^{ 2 }>\frac { { n }^{ 3 } }{ 3 } ,n\in N\) .

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

Prove the following by using the principle of mathematical induction for all n ∊ N \(\frac { 1 }{ 2 } +\frac { 1 }{ 4 } +\frac { 1 }{ 8 } +....+\frac { 1 }{ 2^{ n } } =1-\frac { 1 }{ 2^{ n } } \)

Prove by the principle of mathematical induction that for all n∊N \(\frac { 1 }{ 1.2 } +\frac { 1 }{ 2.3 } +\frac { 1 }{ 3.4 } +...+\frac { 1 }{ n(n+1) } =\frac { n }{ (n+1) } \)