NIMCET 2025 Paper

Reason (simple and clear)
- Elements of $\mathit{A}$ grow exponentially because of $5^{\mathit{n}}$.
- Elements of $\mathit{B}$ grow linearly because $16\left(\mathit{n}-1\right)$.
Exponential expressions rarely equal linear expressions except for very small values of $\mathit{n}$.
Checking small $\mathit{n}$ :
- $\mathit{n}=1$ : $\mathit{A}=5^1-4-1=0,\mathit{B}=16\left(0\right)=0⟶$ match
- $\mathit{n}=2$ : $\mathit{A}=25-8-1=16,\mathit{B}=16\left(1\right)=16⟶$ match
- $\mathit{n}=3$ : $\mathit{A}=125-12-1=112,\mathit{B}=16\left(2\right)=32⟶$ no match
- For larger $\mathit{n},5^{\mathit{n}}$ grows too fast → no further matches.
Thus, the intersection contains only a finite number of elements (in fact exactly two).
So the correct conclusion:
$\mathit{A}\cap \mathit{B}$ is a finite set