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

If x and y are any two distinct integers, then prove by mathematical induction that \(\left( { x }^{ n }-{ y }^{ n } \right) \) is divisible by (x-y), for all \(n\in N\)

Prove that (2n+7)<(n+3)², for all natural numbers n.

Prove that 2n<(n+2)! for all natural numbers 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:\(\left( 1+\frac { 3 }{ 1 } \right) \left( 1+\frac { 5 }{ 4 } \right) \left( 1+\frac { 7 }{ 9 } \right) ...\left( 1+\frac { (2n+1) }{ { n }^{ 2 } } \right) =(n+1)^{ 2 }\)

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

Prove the following by using the principle of mathematical induction for all n ∊ N: 3²ⁿ⁺¹-8n-9 is divisible by 8.

Prove the following by using the principle of mathematical induction for all n∊ N 41ⁿ-14ⁿ is a multiple of 27

3ⁿ>2ⁿ for all n ∊ N.

Prove by the principle of mathematical induction that sinx+sin 2x+sin 3x+...+sin nx =\(\frac { sin\left( \frac { n+1 }{ 2 } \right) x\quad sin\frac { nx }{ 2 } }{ sin\frac { x }{ 2 } } \)for all n∊N