Cinema 4

In the cinema are 1656 seats and in the last row are 105 seats, in each next row 3 seats less. How many are the total rows in the cinema?

Correct answer:

n =  23

Step-by-step explanation:

$$\begin{gathered} s=1656 \\ {a}_1=105 \\ d=-3 \\ {s}_n=n(a_1+a_n)/2=n(a_1+a_1+(n-1)d)/2 \\ 1656=n/2(2\cdot \cdot 105-3\cdot \cdot (n-1)) \\ 3312=n(213-3n) \\ 3312=213n-3n^2 \\ 3n^2-213n+3312=0 \\ 3n^2-213n+3312=0 \\ 3...\text{ prime number} \\ 213=3\cdot 71 \\ 3312=2^4\cdot 3^2\cdot 23 \\ \text{GCD}(3,213,3312)=3 \\ {n}^2-71n+1104=0 \\ a=1;b=-71;c=1104 \\ D=b^2-4ac=71^2-4\cdot 1\cdot 1104=625 \\ D>0 \\ {n}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{71\pm \sqrt{625}}{2} \\ {n}_{1,2}=\frac{71\pm 25}{2} \\ {n}_{1,2}=35.5\pm 12.5 \\ {n}_1=48 \\ {n}_2=23 \\ n=n_2=23 \\ \text{ Verifying Solution: } \\ 1:a(1)=105;S(1)=105 \\ 2:a(2)=102;S(2)=207 \\ 3:a(3)=99;S(3)=306 \\ 4:a(4)=96;S(4)=402 \\ 5:a(5)=93;S(5)=495 \\ 6:a(6)=90;S(6)=585 \\ 7:a(7)=87;S(7)=672 \\ 8:a(8)=84;S(8)=756 \\ 9:a(9)=81;S(9)=837 \\ 10:a(10)=78;S(10)=915 \\ 11:a(11)=75;S(11)=990 \\ 12:a(12)=72;S(12)=1062 \\ 13:a(13)=69;S(13)=1131 \\ 14:a(14)=66;S(14)=1197 \\ 15:a(15)=63;S(15)=1260 \\ 16:a(16)=60;S(16)=1320 \\ 17:a(17)=57;S(17)=1377 \\ 18:a(18)=54;S(18)=1431 \\ 19:a(19)=51;S(19)=1482 \\ 20:a(20)=48;S(20)=1530 \\ 21:a(21)=45;S(21)=1575 \\ 22:a(22)=42;S(22)=1617 \\ 23:a(23)=39;S(23)=1656 \end{gathered}$$

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