Lecture 2 : Revision

Prove that 2n > n ∀positive integers n.

Prove 11ⁿ⁺² + 12²ⁿ⁺¹ is divisible by 133.

Prove that 12ⁿ + 25^n-1 is divisible by 13

The sum of the cubes of three consecutive natural no. is divisible by 9.

Prove (x²ⁿ-1) is divisible by (x-1).

Prove that 3²ⁿ⁺² – 8n – 9 is divisible by 8

Show that the sum of the first n odd natural no is n²..

Show that 2³ⁿ – 1 is divisible by 7 

Prove that n (n + 1) (2n + 1) is divisible by 6.

Prove that x²ⁿ – y²ⁿ is divisible by x + y.

Prove by PMI (ab)ⁿ = aⁿ bⁿ

 Using induction, prove that 10ⁿ + 3.4ⁿ⁺² + 5 is divisible by 9 ∀ n∈N 

Prove that 41ⁿ – 14ⁿ is a multiple of 27 

Prove that 2.7ⁿ + 3.5ⁿ – 5 is divisible by 24 ∀ n ∈ N 

Prove 1.2+2.2²+3.2³+…+n.2ⁿ = (n-1)2 ^{n + 1}+2

Prove (2n+7)<(n+3)²

Prove that 10^2n-1 +1 is divisible by 11

Prove that n(n +1)(n + 5) is multiple of 3.

For every integer n, prove that 7ⁿ - 3ⁿis divisible by 4.