$$
\begin{array}{l}
P=\left\{n \left\lvert\, \frac{3^{n}+4^{n}}{5} \in \mathbf{N}\right., n \in \mathbf{N}\right\}, \\
Q=\left\{m \mid m=(2 k-1)^{2}+1, k \in \mathbf{N}\right\} .
\end{array}
$$

Then the relationship between $P$ and $Q$ is $($.
(A) $P=Q$
(B) $P \varsubsetneqq Q$
(C) $Q \subsetneq P$
(D) $P \not Q$ and $Q \not P$
$-1 .(\mathrm{C})$.
List the unit digits of $3^{n}$ and $4^{n}$, it is easy to see that when $n=2,6,10,14$, $18, \cdots$, $\frac{3^{n}+4^{n}}{5} \in \mathbb{N}$.
Therefore, $P=\left\{n \mid n=4 m^{\prime}-2, m^{\prime} \in \mathbb{N}_{+}\right\}$.
On the other hand, for any $m \in Q, m=4\left(k^{2}-k+1\right)-2$ $\in P$, and $k^{2}-k+1=k(k-1)+1$ is odd, so there are no elements of the form $8 k-2\left(k \in \mathbb{N}_{+}\right)$ in $Q$. But $8 k-2 \in P$, which indicates $Q \subsetneq P$.