# Cinema

Cinema auditorium is built for 3300 people. The first row is planned for 36 seats and each next gradually 4 more. How many rows of seats will have auditorium?

Correct result:

n =  33

#### Solution:

$3300=\dfrac{ n }{ 2 } (a_{1}+an)=\dfrac{ n }{ 2 } (a_{1}+a_{1} + d(n-1)) \ \\ 3300=\dfrac{ n }{ 2 } (a_{1}+a_{1} + d(n-1)) \ \\ 3300=36*n + n/2 * (n-1) *4 \ \\ \ \\ 3300=36 \cdot \ n + n/2 \cdot \ (n-1) \cdot \ 4 \ \\ -2n^2 -34n +3300=0 \ \\ 2n^2 +34n -3300=0 \ \\ \ \\ a=2; b=34; c=-3300 \ \\ D=b^2 - 4ac=34^2 - 4\cdot 2 \cdot (-3300)=27556 \ \\ D>0 \ \\ \ \\ n_{1,2}=\dfrac{ -b \pm \sqrt{ D } }{ 2a }=\dfrac{ -34 \pm \sqrt{ 27556 } }{ 4 } \ \\ n_{1,2}=\dfrac{ -34 \pm 166 }{ 4 } \ \\ n_{1,2}=-8.5 \pm 41.5 \ \\ n_{1}=33 \ \\ n_{2}=-50 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ 2 (n -33) (n +50)=0 \ \\ n>0 \ \\ n=n_{1}=33$

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