Chapter 12.1, Problem 3CFU

a.

To determine

To describe: The value $n=-24.58$ not a valid answer for number of houses sold.

Answer to Problem 3CFU

The number of houses sold cannot be a negative integer.

Explanation of Solution

Given information:

The two possible values for number of houses sold are $n=11\text{ and }n=-24.58$ .

Consider the situation that two possible values for number of houses sold are $n=11\text{ and }n=-24.58$ .

Since, n represent the number of houses sold in order to get the commission so it has to be positive integer.

Number of houses cannot be negative so the value $n=-24.58$ is rejected.

b.

To determine

To calculate: The commissioned earned by her if she sell 10 houses.

Answer to Problem 3CFU

The commissioned earned by her if she sell 10 housesis $\$60000$ .

Explanation of Solution

Given information:

The first commission is $a_1=\$3750$ and common difference is $d=500$ .

Formula used:

A sequence with first term as $a_1$ and next term written as sum of preceding term and common difference then it is called an arithmetic sequence. It is denoted as $a_1,a_1+d,a_1+2d+\dots$ .

The sum of n terms of the arithmetic sequence is $S_n=\frac{n}{2}\left[2a_1+\left(n-1\right)d\right]$ .

Calculation:

Consider the provided information that first commission is $a_1=\$3750$ and common difference is $d=500$ .

The commissioned earned by her if she sell 10 houses $S_{10}$ .

Recall that a sequence with first term as $a_1$ and next term written as sum of preceding term and common difference then it is called an arithmetic sequence. It is denoted as $a_1,a_1+d,a_1+2d+\dots$ .

The sum of n terms of the arithmetic sequence is $S_n=\frac{n}{2}\left[2a_1+\left(n-1\right)d\right]$ or $S_n=\frac{n}{2}\left[a_1+a_n\right]$

Here $a_1=\$3750,d=500\text{ and }n=10$ .

Again apply the sum of n terms of the arithmetic sequence $S_n=\frac{n}{2}\left[2a_1+\left(n-1\right)d\right]$ ,

  $\begin{array}{c}{S}_{10}=\frac{10}{2}\left[2\left(3750\right)+\left(10-1\right)500\right]\\ =\frac{10}{2}\left[7500+4500\right]\\ =60000\end{array}$

Thus, the commissioned earned by her if she sell 10 houses is $\$60000$ .