Prove: 52n-1 is divisible by 24 for all n N
Solution: (a) Proof by induction:
For n=1, we have 52n - 1 = 24.
Next, assume that the result holds for n=k.
For n=k+1, we have the following expression.
52(k+1) - 1
= (52k)(52) - 1
= (52k)(25) - 25 + 25 - 1
= (52k - 1)(25) + 24
By the induction hypothesis, 24 is a factor of 52k - 1, and so it is follows that
24 is a factor of 52n - 1.