6.1 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 - 0.5 points more than for the previous one. How many times did the shooter hit the target, receiving 7 penalty points?
6.1 Let $x$ be the number of misses in a series of 25 shots. Then, using formula (6.3), we get the equation

$$
\frac{2 \cdot 1+0.5(x-1)}{2} x=7, \text{ or } x^{2}+3 x-28=0
$$

from which $x_{1}=4, x_{2}=-7$ (does not fit). Therefore, the shooter hit the target 21 times.

Answer: 21 times.