WBJEE · Maths · Mathematical Induction

Let \(P(n)=3^{2 n+1}+2^{n+2}\) where \(n \in \mathbb{N}\). Then

A \(\mathrm{P}(\mathrm{n})\) is not divisible by any prime integer.
B there exists prime integer which divides \(P(n)\).
C \(P(n)\) is divisible by 5 for all \(n \in \mathbb{N}\).
D \(\mathrm{P}(\mathrm{n})\) is divisible by 3 for all \(\mathrm{n} \in \mathbb{N}\).

Verified Solution

Answer & Solution

Correct Answer

(B) there exists prime integer which divides \(P(n)\).

Step-by-step Solution

Detailed explanation

Hint : \(P(1)=3^3+2^3\) which is divisible by \((3+2)=5\) which is prime \(\therefore\) There exists prime integer which divides \(\mathrm{P}(\mathrm{n})\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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