Chapter 9.4, Problem 33PPSE

Solution

To explain how to find the number of seats in each row using the explicit formula, the number of seats in the 16^th row, and the sum of the seats in 16 rows.

Given information:

There are 16 rows in the meeting room and the first row has 12 seats, and the number of seats increases by 2 with each successive row.

Calculation:

The sequence is an arithmetic sequence with $a_1=12$ and $d=2$ .

Find the nth term of the sequence.

  $\begin{array}{c}{a}_n=a_1+\left(n-1\right)d\\ =12+\left(n-1\right)2\\ =2n+10\end{array}$

Therefore, the number of seats in each row can be found using the explicit formula, $a_n=2n+10$ .

Find the number of seats in the 16^th row.

  $\begin{array}{c}{a}_{16}=2\cdot 16+10\\ =32+10\\ =42\end{array}$

Therefore, there are 42 seats in the 16^th row.

Now, find the total number of seats in 16 rows.

Find the total number of seats in the meeting room by evaluating the sum of the corresponding arithmetic series using the formula, $S_n=\frac{n}{2}\left(a_1+a_n\right)$ .

  $\begin{array}{c}{S}_{16}=\frac{16}{2}\left(a_1+a_{16}\right)\\ =\frac{16}{2}\left(12+42\right)\\ =8\left(54\right)\\ =432\end{array}$

Therefore, there are total 432 seats in the meeting root.