Chapter 9, Problem 9CA

a.

To determine

To write rules giving the salaries $a_n$ and $b_n$ for your $n\text{th}$ year of employment represented by each rule is arithmetic, geometric, or neither.

Answer to Problem 9CA

  $a_n=1000n+19000$

and

  $b_n=20,000\times {(1.04)}^{n-1}$

Explanation of Solution

Given:

Company A and Company B offer the same staring annual salary of $\$20,000.$ Company A gives a $\$1000$ raise each year. Company B gives a $\%4$ raise each year

Calculation:

Writing the arithmetic sequence $a_n$ as follows-

  $20,000,21,000,22,\mathrm{000....}$

  $\begin{array}{l}a=20,000\\ d=1000\end{array}$

Now, computing the $n\text{th}$ term as follows-

  $\begin{array}{l}{a}_n=a+(n-1)d\\ a_n=20,000+(n-1)\cdot 1000\\ a_n=1000n+19000\end{array}$

Writing the geometric sequence $b_n$ as follows-

  $\begin{array}{l}{b}_1=20,000\\ r=1+0.04=1.04\end{array}$

Now, computing the $n\text{th}$ term as follows-

  $\begin{array}{l}{b}_n=b_1{r}^{n-1}\\ $ b_n=20,000\times {(1.04)}^{n-1}$\end{array}$

b.

To determine

To graph each sequence in the same coordinate plane.

Explanation of Solution

Given:

Company A and Company B offer the same staring annual salary of $\$20,000.$ Company A gives a $\$1000$ raise each year. Company B gives a $\%4$ raise each year

Calculation:

Graphing both the equations as below-

  

c.

To determine

To find under what conditions would you choose to work for Company $B$

Explanation of Solution

Given:

Company A and Company B offer the same staring annual salary of $\$20,000.$ Company A gives a $\$1000$ raise each year. Company B gives a $\%4$ raise each year

Calculation:

From the graph it can be noticed that the earnings at company B are greater than the ones at company A after $13$ years of employment. The more you stay in company B, the better you earn as compared to company A, thus you should choose company B if you plan to stay there more than $13$ years.

d.

To determine

To compare the total earnings after $20$ years of employment.

Answer to Problem 9CA

  $5,500$

Explanation of Solution

Given:

Company A and Company B offer the same staring annual salary of $\$20,000.$ Company A gives a $\$1000$ raise each year. Company B gives a $\%4$ raise each year

Calculation:

Computing the $S_{20}$ in each case as follows-

Total earning in company A

  $\begin{array}{l}{S}_{20}=20\left(\frac{a_1+a_{20}}{2}\right)\\ =20\left(\frac{20,000+20,000+19,000}{2}\right)\\ =590,000\end{array}$

Total earning in company B

  $\begin{array}{l}{S}_{20}=20,000\left(\frac{1-{1.04}^{20}}{1-1.04}\right)\\ =20,000\left(\frac{-1.1911}{-0.04}\right)\\ \approx 595,550\end{array}$

Difference in earing is $595,550-590,000=5,500$ .