An auditorium has 40 rows. the first row has 20 seats and each row after has 2 additional seats. How many seats are there total?

This is an arithmetic sequence of the form:

a (n) = a+d (n-1), a=initial term, d=common difference, n=term number

The sum of an arithmetic sequence is the average of the first and last terms times the number of terms.

s (n) = (a+a+d (n-1)) (n/2)

s (n) = (2a+dn-d) (n/2)

s (n) = (2an+dn^2-dn) / 2

We are told that the initial term a=20 and that the common difference, d=2 so:

s (n) = (40n+2n^2-2n) / 2

s (n) = (2n^2+38n) / 2

s (n) = n^2+19n, so for 4o rows the total number of seats will be:

s (40) = 40^2+19 (40)

s (40) = 1600+760

s (40) = 2360