Exercise 33 Page 592

Let's find the total number of seats by following three steps.

1. Write an explicit formula.
2. Find the last term.
3. Find the total number of seats.

Let's do it!

Explicit Formula

Since the number of seats in each row increases by a constant rate, they form an arithmetic sequence. To find the explicit formula we need to know the common difference d and the first term a. a_n=a+(n-1)d Because the number of seats in each row increases by 2, we know that the common difference is d= 2. Since the first row has 12 seats, we have that a= 12. Let's substitute these values in the above formula and simplify.

The explicit formula for the arithmetic sequence formed by the number of seats in each row is a_n=10+2n.

Last Term

We are told that there are 16 rows. Let's substitute n=16 into our explicit formula to find the number of seats in the last row.

The number of seats in the 16th and last row is 42.

Total Seats

Finally, to find the total number of seats we will use the formula for the sum of a finite arithmetic series. S_n=n/2(a_1+a_n) Let's substitute n= 16, a_1= 12, and a_(16) = 42 into the formula.

There are 432 seats in the meeting room.