4.003. In a shooting competition, for each miss in a series of 25 shots, the shooter received penalty points: for the first miss - one penalty point, and for each subsequent miss - half a point more than for the previous one. How many times did the shooter hit the target, having received 7 penalty points?
Solution.

Let $a_{1}=1$ be the first term of the arithmetic progression, $d=\frac{1}{2}$ be its common difference, and $S_{n}=7$ be the sum of the first $n$ terms of this progression, where $n$ is the number of terms. Using formula (4.5), we have

$$
\frac{2+(n-1) \cdot \frac{1}{2}}{2} \cdot n=7, n^{2}+3 n-28=0
$$

from which $n_{1}=-7$ (not suitable); $n_{2}=4$. Therefore: the shooter hit the target 21 times.

Answer: 21 times.