Prove the following by the principle of mathematical induction:
52n – 1 is divisible by 24 for all n ϵ N
Let P(n): 52n - 1 is divisible by 24
Let’s check For n =1
P(1): 52 - 1 = 25 - 1
= 24
Since, it is divisible by 24
So, P(n) is true for n=1
Now, for n=k
52k - 1 is divisible by 24
P(k): 52k - 1 = 24λ - - - - - - - (1)
We have to show that,
52k + 1 - 1 is divisible by 24
52(k + 1) - 1 = 24μ
Now,
52(k + 1) - 1
= 52k.52 - 1
= 25.52k - 1
= 25.(24λ + 1) - 1 using equation (1)
= 25.24λ + 24
= 24λ
Therefore, P(n) is true for n = k + 1
Hence, P(n) is true for all n∈N by PMI