## Task B-4.4.

For every natural number $n$, the sum of the first $n$ terms of a certain sequence is $S_{n}=2.5 n^{2}-4.5 n$.

Which terms of this sequence take values between 10085 and 10095?
## Solution.

The general term of the sequence can be calculated as follows

$$
\begin{array}{rlrl}
a_{n} & =S_{n}-S_{n-1} & 1 \text { point } \\
& =\left(2.5 n^{2}-4.5 n\right)-\left(2.5(n-1)^{2}-4.5(n-1)\right) & \\
& =5 n-7 & & 1 \text { point }
\end{array}
$$

Determine $n$ from the condition $10085<a_{n}<10095$.

$$
\begin{gathered}
10085<5 n-7<10095 \\
10092<5 n<10102 \\
2018.4<n<2020.4
\end{gathered}
$$

The only natural numbers in this interval are

$$
n=2019 \quad \text{and} \quad n=2020
$$

so the required terms are $a_{2019}=5 \cdot 2019-7=10088$ and $a_{2020}=5 \cdot 2020-7=10093$.