15. Let $x>2$ be a real number and $n=2^{2017}$. Calculate the sum of the numbers

$$
\ln x, \ln \sqrt{x}, \ln \sqrt[4]{x}, \ln \sqrt[8]{x}, \ldots, \ln \sqrt[n]{x}
$$
Solution. Since $\frac{\ln x^{2^{s}}}{\frac{1}{\frac{1}{2-1}}}=\frac{1}{2}$ for every $s \in\{1,2, \ldots, 2017\}$, it follows that the given

numbers form a geometric progression with a common ratio of $\frac{1}{2}$. Therefore, the desired sum is

$$
S_{2017}=\ln x \frac{\left(\frac{1}{2}\right)^{2017}-1}{\frac{1}{2}-1}=2\left(\left(\frac{1}{2}\right)^{2017}-1\right) \ln x
$$

2AB. Determine the term with the largest coefficient in the expansion of the binomial

$$
(x+\sqrt{2})^{50}
$$