If n ϵ N, then 121ⁿ – 25ⁿ + 1900ⁿ – (-4)ⁿ is divisible by which one of the following?

Question

Question

This question was previously asked in

NDA (Held On: 22 April 2018) Maths Previous Year paper

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Answer 1

Concepts:

Suppose there is a given statement P (n) involving the natural number n such that

The statement is true for n = 1, i.e., P (1) is true, and
If the statement is true for n = k (where k is some positive integer), then the statement is also true for n = k + 1, i.e., truth of P (k) implies the truth of P (k + 1).

Then, P (n) is true for all natural numbers n

Calculation:

Given:

P (n) = 121ⁿ – 25ⁿ + 1900ⁿ – (-4)ⁿ

Now, P (1) = 121¹ – 25¹ + 1900¹ – (-4)¹

⇒ P (1) = 121 – 25 + 1900 + 4

⇒ P (1) = 2000

Therefore we can say that P (n) is divisible by 2000.