## Problem Statement

Calculate the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty} \frac{1+4+7+\ldots+(3 n-2)}{\sqrt{5 n^{4}+n+1}}
$$
## Solution

$$
\begin{aligned}
& \lim _{n \rightarrow \infty} \frac{1+4+7+\ldots+(3 n-2)}{\sqrt{5 n^{4}+n+1}}= \\
& =\lim _{n \rightarrow \infty} \frac{1}{\sqrt{5 n^{4}+n+1}} \cdot \frac{(1+(3 n-2)) n}{2}=\lim _{n \rightarrow \infty} \frac{n(3 n-1)}{2 \sqrt{5 n^{4}+n+1}}= \\
& =\lim _{n \rightarrow \infty} \frac{\frac{1}{n^{2}} n(3 n-1)}{\frac{1}{n^{2}} 2 \sqrt{5 n^{4}+n+1}}=\lim _{n \rightarrow \infty} \frac{3-\frac{1}{n}}{2 \sqrt{5+\frac{1}{n^{3}}+\frac{1}{n^{4}}}}= \\
& =\frac{3-0}{2 \sqrt{5+0+0}}=\frac{3}{2 \sqrt{5}}
\end{aligned}
$$

## Problem Kuznetsov Limits 6-8