## Task B-3.3.

Calculate

$$
\frac{1}{\log _{2} n^{2}}+\frac{1}{\log _{\frac{3}{2}} n^{2}}+\frac{1}{\log _{\frac{4}{3}} n^{2}}+\ldots+\frac{1}{\log _{\frac{n}{n-1}} n^{2}}, \quad n \in \mathbb{N}
$$
## Solution.

$$
\begin{aligned}
& \frac{1}{\log _{2} n^{2}}+\frac{1}{\log _{\frac{3}{2}} n^{2}}+\frac{1}{\log _{\frac{4}{3}} n^{2}}+\ldots+\frac{1}{\log _{\frac{n}{n-1}} n^{2}}= \\
& =\log _{n^{2}} 2+\log _{n^{2}} \frac{3}{2}+\log _{n^{2}} \frac{4}{3}+\cdots+\log _{n^{2}} \frac{n}{n-1}= \\
& =\log _{n^{2}}\left(2 \cdot \frac{3}{2} \cdot \frac{4}{3} \cdots \frac{n}{n-1}\right)= \\
& =\log _{n^{2}} n=\frac{1}{2}
\end{aligned}
$$

## Note:

The problem can also be solved by expressing the given logarithms using logarithms with base 10 (or any other base) and reducing the given expression to a common denominator.