Chapter 4, Problem 9CT

a.

To determine

To Calculate : To calculate the number of seats in ROW 25.

Answer to Problem 9CT

The number of seats in ROW 25 is $=114$ .

Explanation of Solution

Given information : First row contain $42$ seats and each row after the first has three more seats than the row before it.

According to the question, we get an arithmetic sequence of number of seats in each row as follow-

Number of seats in first row $=42$ .

Number of seats in second row is three more $=45$ .

Number of seats in third row is three more $=48$ .

and so on.

Hence, we get the sequence $42,45,48,51,54,\mathrm{...}$

Then, n^th term of this sequence will be

  $\begin{array}{c}{a}_n=42+3\left(n-1\right)\\ =42+3n-3\\ =39+3n\end{array}$

Therefore, to calculate the number of seats in ROW 25, we need to calculate the 25^th term of the above sequence,

  $\begin{array}{c}{a}_{25}=39+3\times 25\\ =39+75\\ =114\end{array}$

Thus, the number of seats in ROW 25is $=114$ .

b.

To determine

To determine the row which contain $90$ seats.

Answer to Problem 9CT

The ROW 17 contain $90$ seats.

Explanation of Solution

Given information : First row contain $42$ seats and each row after the first has three more seats than the row before it.

According to the question, we get an arithmetic sequence of number of seats in each row as follow-

Number of seats in first row $=42$ .

Number of seats in second row is three more $=45$ .

Number of seats in third row is three more $=48$ .

and so on.

Hence, we get the sequence $42,45,48,51,54,\mathrm{...}$

Then, n^th term of this sequence will be

  $\begin{array}{c}{a}_n=42+3\left(n-1\right)\\ =42+3n-3\\ =39+3n\end{array}$

Therefore, to determine the number of row that contain $90$ seats, we need to solve the equation,

  $\begin{array}{c}90=39+3n\\ 3n=90-39\\ 3n=51\\ n=\frac{51}{3}\\ n=17\end{array}$

Thus, the ROW 17 contains $90$ seats.