3-1. Calculate the sum

$$
S=\frac{2014}{2 \cdot 5}+\frac{2014}{5 \cdot 8}+\frac{2014}{8 \cdot 11}+\ldots+\frac{2014}{2012 \cdot 2015}
$$

In your answer, specify the remainder from dividing by 5 the even number closest to the obtained value of $S$.
Solution. Since

$$
\begin{aligned}
& \frac{1}{2 \cdot 5}+\frac{1}{5 \cdot 8}+\frac{1}{8 \cdot 11}+\ldots+\frac{1}{2012 \cdot 2015}= \\
& =\frac{1}{3} \cdot\left(\left(\frac{1}{2}-\frac{1}{5}\right)+\left(\frac{1}{5}-\frac{1}{8}\right)+\left(\frac{1}{8}-\frac{1}{11}\right)+\ldots\left(\frac{1}{2012}-\frac{1}{2015}\right)\right)= \\
& \quad=\frac{1}{3} \cdot\left(\frac{1}{2}-\frac{1}{2015}\right)=\frac{1}{6} \cdot \frac{2013}{2015}
\end{aligned}
$$

the desired sum is

$$
\frac{2014 \cdot 2013}{6 \cdot 2015}=335.33 \ldots
$$

The desired natural number is 336.

Answer: 1. (V)

Options.

![](https://cdn.mathpix.com/cropped/2024_05_06_997c53c06135e2de9c16g-03.jpg?height=65&width=635&top_left_y=924&top_left_x=762)