Chapter 8.2, Problem 63E

To determine

To calculate: The number of seats in the front row.

Answer to Problem 63E

There are $m=x-\left[\frac{nx-y}{0.5n}\right]$ seats in the first row.

Explanation of Solution

Given information:

A theater has n rows of seats, and each row has d more seats than the row in front of it. There are x seats in the last $(nth)$ row and a total of y seats in the entire theater.

Formula used:

Arithmetic difference − the difference of consecutive term is constant. The constant difference is called the common difference and is denoted by d.

Therefore,

  $d=a_{n+1}-a_n;n\ge 1$ .

The $nth$ tem of an arithmetic sequence with first term $a_1$ and common difference $d$ is given by : $a_n=a_1+(n-1)d$ .

The sum of the first n terms of arithmetic series is

  $S_n=n\left(\frac{a_1+a_n}{2}\right).$

Calculation:

Let the number of seats in front row of the theatre are m .

Let the number of seats in back row of the theatre are x .

Each row has d more seats than the last.

Therefore,

  $\begin{array}{l}=a_n=a_1+(n-1)d\text{ (i)}\\ =x=m+(n-1)d\text{ (ii)}\end{array}$

The total number of seats in theater are y.

Now,

  $y={\displaystyle \sum _{i=0}^{n-1}m+id}\text{ (iii)}$

Sum of n terms in A.P is −

  $\begin{array}{l}y=\frac{n}{2}\left[2a+(n-1)d\right]\\ y=nm+\frac{n(n-1)d}{2}\\ y=nm+0.5dn(n-1)\text{ (iv)}\end{array}$

Taking value of m from equation (ii) and substituting it in equation (iv) we get,

  $x-(n-1)d=m$ and

  $\begin{array}{l}y=n\left[x-(n-1)d\right]+0.5dn(n-1)\\ =nx-n(n-1)d+0.5dn(n-1)\\ =nx+n(n-1)d\left[-1+0.5\right]\\ =nx-0.5nd(n-1)\end{array}$

Now, the value of d can come out from the above equation.

  $\begin{array}{l}=y=nx-0.5nd(n-1)\\ =y-nx=-0.5nd(n-1)\\ =\frac{y-nx}{-0.5n(n-1)}=d\\ =\frac{nx-y}{0.5n(n-1)}=d\end{array}$

Substituting the value of d in equation (ii) we get,

  $\begin{array}{l}=x=m+(n-1)\left[\frac{nx-y}{0.5n(n-1)}\right]\\ =x=m+\left[\frac{nx-y}{0.5n}\right]\\ =m=x-\left[\frac{nx-y}{0.5n}\right]\end{array}$

Hence , the number of seats in the front row are $m=x-\left[\frac{nx-y}{0.5n}\right]$ .