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.

3

(a)

Prove by the principle of mathematical induction that \({ 3 }^{ n }>{ 2 }^{ n }\) , for all \(n\in N\) .

3

(a)

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

3

(a)

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

3

(a)

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

3

(a)

Prove that \(1+2+3......n<\frac { 1 }{ 8 } { (2n+1) }^{ 2 }\) , for all natural numbers n.

3

(a)

Prove by mathematical induction that \({ 1 }^{ 3 }+{ 2 }^{ 3 }+{ 3 }^{ 3 }+...+{ n }^{ 3 }=\left[ \frac { n(n+1) }{ 2 } \right] ^{ 2 }\), for all \(n\varepsilon N\)

3

(a)

Prove the rule of exponents (ab)ⁿ=aⁿ bⁿ  by using principle of mathematical induction fo every natural number.

3

(a)

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.

3

(a)

Let P(n) be the statement "3ⁿ">n". If p(n) is true, prove that p(n+1) is true.

3

(a)