(2) Given $K \in \mathrm{N}$ and experiment $a>0$, under the following conditions
$$
k_{1}+k_{2}+\cdots+k_{r}=k, k_{i} \in \mathbf{N}, 1 \leqslant r \leqslant k
$$

find the maximum value of $a^{k_{1}}+a^{k_{2}}+\cdots+a^{k_{r}}$.
Let $k_{1}, k_{2} \in \mathbf{N}$, then
$$
a^{k_{1}+k_{2}-1}+a-a^{k_{1}}-a^{k_{2}}=a\left(a^{k_{1}-1}-1\right)\left(a^{k_{2}-1}-1\right) \geqslant 0
$$
(when $a>1$, $a^{k_{1}-1}-1 \geqslant 0, a^{k_{2}-1}-1 \geqslant 0$; when $0<a \leqslant 1$, $a^{k_{1}-1}-1 \leqslant 0, a^{k_{2}-1}-1 \leqslant 0$), hence
$$
a^{k_{1}}+a^{k_{2}} \leqslant a+a^{k_{1}+k_{2}-1}
$$

Therefore, $a^{k_{1}}+a^{k_{2}}+\cdots+a^{k} \leqslant a+a^{k_{1}+k_{2}-1}+a^{k_{3}}+\cdots+a^{k_{r}} \leqslant$
$$
\begin{array}{l}
2 a+a^{k_{1}+k_{2}+k_{3}-2}+a^{k_{4}}+\cdots+ \\
a^{k} \leqslant \cdots \leqslant(r-1) a+a^{k-r+1}
\end{array}
$$

And when $k_{1}=k_{2}=\cdots=k_{r-1}=1, k_{r}=k-r+1$, the value $(r-1) a+a^{k-r+1}$ is achieved. Therefore, the maximum value is
$$
(r-1) a+a^{k-r+1}
$$