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

2

(a)

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

2

(a)

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

2

(a)

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

2

(a)

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

2

(a)

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

2

(a)

Prove that 2²ⁿ -1 is divisible by 3, for all natural numbers n.

2

(a)

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

2

(a)

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

2

(a)