Answer
20 rows
Work Step by Step
The number of seats in a row form an arithmetic sequence with
first term $a=15$
and common difference $d=3$.
The sum of the first n terms (total seats in n rows) is
$S_{n}=\displaystyle \frac{n}{2}[2a+(n-1)d]$
Given $S_{n}=870$, we find n.
$870=\displaystyle \frac{n}{2}[2(15)+(n-1)3]$
$870=\displaystyle \frac{n}{2}(27+3n) \qquad/\times 2$
$1740=3n^{2}+27n$
$3n^{2}+27n-1740=0 \qquad/\div 3$
$n^{2}+9n-580=0$
Quadratic formula:
$n=\displaystyle \frac{-9\pm\sqrt{81+2320}}{2}=\frac{-9\pm 49}{2}$
...discarding the negative solution,
$n=\displaystyle \frac{-9+49}{2}=20$
The theater has 20 rows.
