If p(n): "49ⁿ +16ⁿ +k is divisible by 64 for n\(\epsilon \)N" is true, then find the least negative integral value of K.

2

(a)

If P(n): "3.5²ⁿ⁺¹ +2³ⁿ⁺¹ is divisible by  for all n \(\in\) N" is true, then find the value of \(\lambda \).

2

(a)

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

2

(a)

Prove that 1+2+2²+...+2ⁿ = 2ⁿ⁺¹ 1 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)