precal question

Question

An architect designs a theater with 18 seats in the first row, 22 in the second, 26 in the third, and so on. If the theater is to have a seating capacity of 1218, how many rows must the architect use in his design?

 rows

Answer 1

Hi Viviana,

The formula for the n^th term of an arithmetic sequence is: a_n = a₁ + (n - 1)*d.

The formula for the arithmetic sum is S_n = n(a₁ + a_n)/2.

Substituting the arithmetic sequence a_n into S_n:

S_n = n(a₁ + [a₁ + (n - 1)*d])/2

With a₁ = 18; d = 4; S_n = 1218

1218 = n(18 + [18 +(n - 1)*4])/2

1218 = n(36 + 4n - 4)/2

1218 = 2n² + 16n

2n² + 16n -1218 = 0, you can solve this.

n = 21 and -29

The architect will need 21 rows.

I hope this helps, Joe.