Prove that 2+6+18+.....2.3^n-1 = (3ⁿ - 1) for all \(n\in N\).

Prove that 1+2+2²+...+2ⁿ = 2ⁿ⁺¹ 1 for all natural numbers n.

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

Using principle of mathematical induction, prove that
\(\frac { 1 }{ 2.5 } +\frac { 1 }{ 5.8 } +\frac { 1 }{ 8.11 } +....+\frac { 1 }{ (3n-1)(3n+2) } =\frac { n }{ 6n+4 } \)

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 \(2n+1 <{ 2 }^{ n }\),for all natural numbers \(n(n\ge 3)\) by using principle of mathematical induction.

Prove by principle of mathematical induction that, the sum of first n natural numbers is \(\frac { n(n+1) }{ 2 } \)

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

Prove by the principle of mathematical induction that \(1+3+3^{ 2 }+....+3^{ n-1 }=\frac { 3^{ n }-1 }{ 2 } \) , for all \(n \in N\).
 

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.

Prove that \(1.4.7+2.5.8+3.6.9+....+n(n+3)(n+6)=\frac { n }{ 4 } (n+1)(n+6)(n+7)\)for all natural numbers 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 by the principle of mathematical induction that, for all \(n\in N\),4ⁿwhen divided by 3, the remainder is always 1.

For all positive integer n, prove that \(\frac { { n }^{ 7 } }{ 7 } +\frac { { n }^{ 5 } }{ 5 } +\frac { 2 }{ 3 } { n }^{ 3 }-\frac { n }{ 105 }\) is an integer.

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