If n is a positive integer, then 52n+2 – 24n – 25 is divisible by(

1.	If n is a positive integer, then 52n+2 – 24n – 25 is divisible by(a) 574 (b) 576 (c) 675 (d) 575
Answer» Answer: (B) 576 For n = 1, 5^{2n + 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: 5^{2n + 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) = 5^{2k + 2} – 24k – 25 is divisible by 576 is true, i.e., 5^2k+2 – 24k – 25 = 576m, m∈N ....(i) ∴ T(k + 1) = 5^{2(k + 1)+2} – 24 (k + 1) – 25 = 5^{2k + 2} . 25 – 24k – 24 – 25 = 5^{2k + 2} . 25 – 24k – 49 = 25 (5^{2k + 2} – 24k – 25) + 24. (24k) + 576 = 25. (576m) + 576k + 576 (From (i)) = 576 (25m + k + 1) ⇒ 2^{2(k + 1) + 2} – 24 (k + 1) – 25 is divisible by 576 ⇒ T(k + 1) is true, whenever T(k) is true. ⇒ 5^{2n + 2} – 24k – 25 is divisible by 576 ∀ n∈N

If n is a positive integer, then 52n+2 – 24n – 25 is divisible by(a) 574 (b) 576 (c) 675 (d) 575

Answer»

Answer: (B) 576

For n = 1,

5^{2n + 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: 5^{2n + 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) = 5^{2k + 2} – 24k – 25 is divisible by 576 is true,

i.e., 5^2k+2 – 24k – 25 = 576m, m∈N ....(i)

∴ T(k + 1) = 5^{2(k + 1)+2} – 24 (k + 1) – 25

= 5^{2k + 2} . 25 – 24k – 24 – 25

= 5^{2k + 2} . 25 – 24k – 49

= 25 (5^{2k + 2} – 24k – 25) + 24. (24k) + 576

= 25. (576m) + 576k + 576 (From (i))

= 576 (25m + k + 1)

⇒ 2^{2(k + 1) + 2} – 24 (k + 1) – 25 is divisible by 576

⇒ T(k + 1) is true, whenever T(k) is true.

⇒ 5^{2n + 2} – 24k – 25 is divisible by 576 ∀ n∈N