4. In the first row of the hall, there are 15 seats, and in each subsequent row, there are three more seats.

a) The hall has 870 seats. How many rows does this hall have, and how many seats are in its last row?

b) The ticket for a performance costs 9 euros for children and 12 euros for adults. At the morning performance, the hall was fully occupied by children. What is the minimum number of adults that would need to occupy the hall at the afternoon performance for the hall owner to earn more in the afternoon than in the morning?
4. We find that the number of seats in each row are terms of an arithmetic sequence $a_{1}=$ 15 and $d=3$, the total number of seats is the sum of the arithmetic sequence. We use the formula $s_{n}=\frac{n}{2}\left(2 a_{1}+(n-1) d\right)$. Substituting the data, we get the equation $870=\frac{n}{2}(30+(n-1) \cdot 3)$. Simplifying the equation, we get $n^{2}+9 n-580=0$. The equation has two solutions $n_{1}=20$ and $n_{2}=-29$, where the negative solution is discarded. The number of seats in the last row is $a_{20}=a_{1}+(20-1) d=15+19 \cdot 3=72$. We write the answer: The hall has 20 rows and 72 seats in the last row. We calculate the revenue for a fully occupied hall with children $870 \cdot 9$ euros $=7830$ euros. We solve the inequality $x \cdot 12>7830$. We get $x>652.5$. We write the answer: The hall must be occupied by at least 653 adults.

![](https://cdn.mathpix.com/cropped/2024_06_07_97c791425a4f6cf5fca5g-14.jpg?height=63&width=1642&top_left_y=1847&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_97c791425a4f6cf5fca5g-14.jpg?height=57&width=1630&top_left_y=1899&top_left_x=207)

Solution to the equation $n_{1}=20$ and $n_{2}=-29 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

Calculation of the number of seats in the last row $a_{20}=a_{1}+(20-1) d=15+19 \cdot 3=72 \ldots .1$ point

Written answer . ......................................................................................................................

![](https://cdn.mathpix.com/cropped/2024_06_07_97c791425a4f6cf5fca5g-14.jpg?height=51&width=1627&top_left_y=2102&top_left_x=206)

Written answer . .......................................................................................