1. Find the natural number $n$ if the tenth term in the expansion of the binomial $\left(\frac{1}{5} x+\frac{2}{5}\right)^{n}$ has the largest coefficient.
Solution. The $k$-th term is $a^{k}=\binom{n}{k}\left(\frac{2}{5}\right)^{k} \frac{1}{5^{n-k}}=\binom{n}{k} \frac{2^{k}}{5^{n}}$. Since the tenth term has the largest coefficient, it follows that $a_{8}12.5$ and $n<14$. Therefore, the desired natural number is $n=13$.