Atheater is set up so that there are 20 seats in the first row, and 4 additional seats in each consecutive row to the back of the theater where there are 144 seats. how many rows are there in the theater?

there are needed n=5.10 rows , but since it is not a round number (and for n=5 , S=140 ) , we could hypothesise that there are missing the additional 4 seats the first row , and the consecutive rows contains 4 additional seats than the first one , without counting the addition of 4 into the first row

Explanation:

Since the number of seats increases by 4 for every row , we have the formula

a(n+1) = a(n) + 4

therefore

a(n+1) - a(n) = 4 , summing  a(2) - a(1) ...

a(n) - a(1) = 4*(n-1)

a(n) = a(1) + 4*(n-1)

if in the first row a(1)=20

then the total number of rows S is obtained by summing all the a(n) for all the values of n (number of rows)

S= ∑ a(n) = ∑ [ a(1) + 4*(n-1) ] = ∑ [ (a(1)- 4) + 4*n ] = [a(1)-4]*n + 4*n*(n+1)/2

S= [a(1)-4]*n + 4*n*(n+1)/2 = [a(1)-4]*n + 2*n²+ 2*n = [a(1)-2]*n + 2*n²

thus

2*n² + [a(1)-2]*n - S = 0

replacing values a(1) = 20 seats , S=144 seats

2*n² + 18*n - 144 =0

n = [-18 + √(18² + 4*2*144)]/4 = 5.10

therefore n=5.10 rows are needed

since it is not a round number ( for n=5 , S=140 ) , we could hypothesise that there are missing the additional 4 seats the first row , and the consecutive rows contains 4 additional seats than the first one , without counting the addition of 4 into the first row

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