Chapter 9.3, Problem 39E

Solution

To identify: How many seats is in section J the first row of seating in section J of the Athena Arena has 7 seats. In all, there are 25 rows of seats in section J, each row containing two more seats than the row preceding it.

There are $775$ seat in section j.

Given information:

The given term is. $a_1=7$

Number of row $25$

Explanation:

Consider the given information.

Each row contain two more seats than the preceding row, which implies that the common difference is $d=2$

Put $a_1=7\text{and}d=2$ into $a_n=a_1+\left(n-1\right)d$ .

  $\begin{array}{c}{a}_n=7+\left(n-1\right)2\\ a_n=7+2n-2\\ a_n=2n+5\end{array}$

Put the value of $n=25$ into $a_n=2n+5$

  $\begin{array}{l}{a}_n=2n+5\\ a_{25}=2\left(25\right)+5\\ a_{25}=55\end{array}$

Formula for the sum of the first $nth$ term in an arithmetic sequence: $S_n=\frac{n}{2}\left(a_1+a_n\right)$

Put the value of $a_1=7,n=25\text{and}{a}_{25}=25$

  $\begin{array}{l}{S}_n=\frac{n}{2}\left(a_1+a_n\right)\\ S_n=\frac{25}{2}\left(7+55\right)\\ S_n=\frac{n}{2}\left(62\right)\\ S_n=25\left(31\right)\\ S_n=775\end{array}$

So, there are $775$ seat in section j.