Task B-4.3. How many rational terms are there in the expansion of the binomial $(\sqrt{2013}+\sqrt[3]{2013})^{2012}$?
## Solution.

The general term is given by the formula

$$
\begin{aligned}
& \binom{2012}{k} 2013^{\frac{2012-k}{2}} \cdot 2013^{\frac{k}{3}}= \\
= & \binom{2012}{k} 2013^{1006-\frac{k}{6}}
\end{aligned}
$$

The exponent $1006-\frac{k}{6}$ must be a natural number or 0.

Therefore, $k$ must be of the form $k=6n, 0 \leq k \leq 2012, n \in \mathbb{N}_{0}$.

These are the numbers $0, 6, 12, 18, \ldots, 2010$. They form an arithmetic sequence with a difference of 6. There are 336 of them.