Prove that by using the principle of mathematical induction for all `n in N`:
`1.2+ 2.2^(2)+ 3.2^(3)+ ....+ n.2^(n)= (n-1)2^(n+1)+2`

Prove that by using the principle of mathematical induction for all n in N : 1.3+ 2.3^(2)+ 3.3.^(3)+ ....+ n.3^(n)= ((2n-1)3^(n+1)+3)/(4)

Prove that by using the principle of mathematical induction for all n in N : (2n+7) lt (n+3)^(2)

Prove that by using the principle of mathematical induction for all n in N : 1.2+ 2.3+ 3.4+ .....+n(n+1)= [(n(n+1)(n+2))/(3)]

Prove that by using the principle of mathematical induction for all n in N : 1^(2)+3^(2)+5^(2)+...(2n-1)^(2)= (n(2n-1)(2n+1))/(3)

Prove that by using the principle of mathematical induction for all n in N : 1+ 3+ 3^(2)+ …. + 3^(n-1)= ((3^(n)-1))/(2)

Prove that by using the principle of mathematical induction for all n in N : 10^(2n-1)+1 is divisible by 11

Prove that by using the principle of mathematical induction for all n in N : 41^(n)-14^(n) is multiple of 27

Prove that by using the principle of mathematical induction for all n in N : 1.2.3+ 2.3.4+ ....+ n(n+1)(n+2)=(n(n+1)(n+2)(n+3))/(4)

Prove that by using the principle of mathematical induction for all n in N : 1^(3)+ 2^(3)+3^(3)+ ……+n^(3)= ((n(n+1))/(2))^(2)

Prove that by using the principle of mathematical induction for all n in N : 3^(2n+2)-8n-9 is divisible by 8