Answer: (B) 576
For n = 1,
52n + 2 – 24n – 25 = 54 – 24 – 25 = 625 – 49 = 576 which is divisible by 576 and none of the other given alternative.
∴ To prove: 52n+2 – 24n – 25 is divisible by 576 using mathematical induction.
Let T(n) be the statement: 52n + 2 – 24n – 25 is divisible by 576 ∀ n∈N.
Basic Step:
For n = 1, T(1) = 54 – 24 – 25 = 576 which is divisible by 576.
⇒ T(1) is true.
Induction Step:
Assume T(k) where n = k, k∈N to be true i.e.,
T(k) = 52k + 2 – 24k – 25 is divisible by 576 is true,
i.e., 52k+2 – 24k – 25 = 576m, m∈N ....(i)
∴ T(k + 1) = 52(k + 1)+2 – 24 (k + 1) – 25
= 52k + 2 . 25 – 24k – 24 – 25
= 52k + 2 . 25 – 24k – 49
= 25 (52k + 2 – 24k – 25) + 24. (24k) + 576
= 25. (576m) + 576k + 576 (From (i))
= 576 (25m + k + 1)
⇒ 22(k + 1) + 2 – 24 (k + 1) – 25 is divisible by 576
⇒ T(k + 1) is true, whenever T(k) is true.
⇒ 52n + 2 – 24k – 25 is divisible by 576 ∀ n∈N