Cinema

The 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 an auditorium?

Correct answer:

n =  33

Step-by-step explanation:

$$\begin{gathered} 3300=\frac{n}{2}(a_1+an)=\frac{n}{2}(a_1+a_1+d(n-1)) \\ 3300=\frac{n}{2}(a_1+a_1+d(n-1)) \\ 3300=36\cdot n+n/2\cdot (n-1)\cdot 4 \\ 3300=36\cdot n+n/2\cdot (n-1)\cdot 4 \\ -2n^2-34n+3300=0 \\ 2n^2+34n-3300=0 \\ 2...\text{ prime number} \\ 34=2\cdot 17 \\ 3300=2^2\cdot 3\cdot 5^2\cdot 11 \\ \text{GCD}(2,34,3300)=2 \\ {n}^2+17n-1650=0 \\ a=1;b=17;c=-1650 \\ D=b^2-4ac=17^2-4\cdot 1\cdot (-1650)=6889 \\ D>0 \\ {n}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{-17\pm \sqrt{6889}}{2} \\ {n}_{1,2}=\frac{-17\pm 83}{2} \\ {n}_{1,2}=-8.5\pm 41.5 \\ {n}_1=33 \\ {n}_2=-50 \\ n>0 \\ n=n_1=33 \end{gathered}$$

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