Prove, using induction

Question

Prove, using induction, that 3⁽²ⁿ⁻¹⁾ + 1 is divisible by 4 for all n ≥ 1.

 

Answer 1

3(2n - 1) + 1 = 4m for some m ∈ Z

is it true for n = 1?

3[2(1) - 1] + 1 = 4m
3(2 - 1) + 1 = 4m
31 + 1 = 4m
3 + 1 = 4m
1 = m, 1 ∈ Z ∴ True

if true for n, is it true for (n+1)?
Let n, m, p ∈ Z
if m exists such that 3(2n-1)+ 1 = 4m, then
does p exist, such that
3[2(n+1)-1] + 1 = 4p
3(2n+2-1) + 1 = 4p
(32)[3(2n-1) + 1] = 4p
9 (4m) = 4p
36 m = 4p
9m = p

if m ∈ Z, 9m ∈ Z, p ∈ Z     ∴ True