A theater 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