Question 1: Find $\mathrm{C}_{n}^{0}+\mathrm{C}_{n}^{3}+\mathrm{C}_{n}^{6}+\cdots\left(n \in \mathbf{N}_{+}\right)$.

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By the binomial theorem, we have
$$
(1+x)^{n}=\sum_{k=0}^{n} \mathrm{C}_{n}^{k} x^{k} \text {. }
$$

Substituting $x=1, \omega, \omega^{2}\left(\omega=\mathrm{e}^{\frac{2 \pi x}{3}}\right)$ into the above equation, and adding the resulting three equations on both sides, and noting that
$$
\begin{aligned}
\omega^{3 k}= & 1(k \in \mathbf{N}), \text { and } 1+\dot{\omega}+\omega^{2}=0, \text { we get } \\
& \mathrm{C}_{n}^{0}+\mathrm{C}_{n}^{3}+\mathrm{C}_{n}^{6}+\cdots \\
= & \frac{1}{3}\left[2^{n}+\left(-\omega^{2}\right)^{n}+(-\omega)^{n}\right] \\
= & \frac{1}{3}\left(2^{n}+\mathrm{e}^{\frac{\pi i}{3} n}+\mathrm{e}^{-\frac{\pi i}{3} n}\right) \\
= & \frac{1}{3}\left(2^{n}+2 \cos \frac{n \pi}{3}\right) .
\end{aligned}
$$