Question

Prove by method of induction, for all n ∈ N
$\rm 3^{n} - 2n-1$ is divisible by 4.

Answer 1

Answer:

3ⁿ - 2n  - 1  is divisible by 4

Step-by-step explanation:

p(n) = 3ⁿ - 2n  - 1  is divisible by 4

p(1) = 3¹ - 2*1 - 1 = 3 - 2 - 1  = 0  divisible by 4

p(2) = 3² - 2*2 - 1 = 9 - 4 - 1 = 4  divisible by 4

let assume

p(a) = 3ᵃ - 2a - 1  is divisible by 4

=> 3ᵃ - 2a - 1 = 4K

=> 3ᵃ = 4K + 2a + 1

p(a + 1) = 3ᵃ⁺¹ - 2(a+1) - 1

= 3*3ᵃ - 2a - 2 - 1

= 3(4K + 2a + 1) - 2a - 3

=  12k + 6a + 3 - 2a - 3

= 12K + 4a

= 4(3K + a)

divisible by 4

Hence

3ⁿ - 2n  - 1  is divisible by 4