NIMCET 2025 — Mathematics PYQ

For $n\in \mathbb{N}$,  
$$<br>5^n=(1+4{)}^n=1+4n+\sum _{k=2}^n(\genfrac{}{}{0ex}{}{n}{k})4^k<br>$$

Every term $\left(\genfrac{}{}{0ex}{}{n}{k}\right)4^k$ for $k\ge 2$ is divisible by $16$.  
Hence,  
$$<br>5^n-4n-1\equiv 0(\mathrm{mod}16)<br>$$

This means $5^n-4n-1\in B$ for all $n$.  
Therefore,  
$$<br>A\subset B<br>$$

However, note that $32\in B$, but no value of $n$ satisfies  
$$<br>5^n-4n-1=32<br>$$
(since the possible values are $0,16,112,\dots$).  
Thus, $B\not\subset A$.

$$<br>\boxed{\text{Hence, the correct choice is (b) }A\subset B.}<br>$$