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in Mathematical Induction by (22.8k points)
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If n is a positive integer, then 52n+2 – 24n – 25 is divisible by

(a) 574 

(b) 576 

(c) 675 

(d) 575 

1 Answer

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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

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