Suppose P (n): 52n + 2 – 24n – 25 is divisible by 576
Now let us check for n = 1,
P (1): 52.1 + 2 – 24.1 – 25
: 625 – 49
: 576
P (n) is true for n = 1. Where, P (n) is divisible by 576
Then, let us check for P (n) is true for n = k, and have to prove that P (k + 1) is true.
P (k): 52k + 2 – 24k – 25 is divisible by 576
: 52k + 2 – 24k – 25 = 576λ …. (i)
Now we have to prove,
52k + 4 – 24(k + 1) – 25 is divisible by 576
5(2k + 2) + 2 – 24(k + 1) – 25 = 576μ
Therefore,
= 5(2k + 2) + 2 – 24(k + 1) – 25
= 5(2k + 2).52 – 24k – 24 – 25
= (576λ + 24k + 25)25 – 24k– 49 by using equation (i)
= 25. 576λ + 576k + 576
= 576(25λ + k + 1)
= 576μ
P (n) is true for n = k + 1
Thus, P (n) is true for all n ∈ N.