## Problem Statement

Calculate the limit of the numerical sequence:

$\lim _{n \rightarrow \infty} \frac{n^{2}+\sqrt{n}-1}{2+7+12+\ldots+(5 n-3)}$
## Solution

$\lim _{n \rightarrow \infty} \frac{n^{2}+\sqrt{n}-1}{2+7+12+\ldots+(5 n-3)}=\lim _{n \rightarrow \infty} \frac{n^{2}+\sqrt{n}-1}{\left(\frac{(2+(5 n-3)) n}{2}\right)}=$

$$
\begin{aligned}
& =\lim _{n \rightarrow \infty} \frac{2\left(n^{2}+\sqrt{n}-1\right)}{(5 n-1) n}=\lim _{n \rightarrow \infty} \frac{\frac{1}{n^{2}} 2\left(n^{2}+\sqrt{n}-1\right)}{\frac{1}{n^{2}}(5 n-1) n}= \\
& =\lim _{n \rightarrow \infty} \frac{2\left(1+\sqrt{\frac{1}{n^{3}}}-\frac{1}{n^{2}}\right)}{5-\frac{1}{n}}=\frac{2(1+0-0)}{5-0}=\frac{2}{5}
\end{aligned}
$$

## Problem Kuznetsov Limits 6-22