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.

a_n=a+(n-1)d

SubstituteII

d= 2, a= 12

a_n = 12 + (n-1) 2

▼

Simplify right-hand side

Distr

Distribute 2

a_n = 12 + 2n - 2

SubTerms

Subtract terms

a_n = 10 + 2n

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.

a_n = 10 + 2n

Substitute

n= 16

a_(16) = 10 + 2( 16)

▼

Evaluate right-hand side

Multiply

Multiply

a_(16) = 10 + 32

AddTerms

Add terms

a_(16) = 42

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.

S_n = n/2(a_1+a_n)

Substitute

n= 16

S_(16)= 16/2(a_1+a_(16))

SubstituteII

a_1= 12, a_(16)= 42

S_(16)=16/2( 12+ 42)

▼

Evaluate right-hand side

CalcQuot

Calculate quotient

S_(16)=8(12+42)

AddTerms

Add terms

S_(16)=8(54)

Multiply

Multiply

S_(16)=432

There are 432 seats in the meeting room.