Chapter 12, Problem 29RE

a.

To determine

To find: a formula for the salary of the teacher.

Answer to Problem 29RE

  $A_n=32000{\left(1.05\right)}^{n-1}$

Explanation of Solution

Given information:

The teacher makes $32000 in the first year and gets a raise of 5% each year.

Calculation:

5% is decimal can be written as 0.05.

If the first-year salary is $A_1$ , then

  $\begin{array}{l}{A}_2=A_1+0.05A_1=A_1\left(1+0.05\right)=1.05A_1\\ A_3=A_2+0.05A_2=A_2\left(1+0.05\right)=1.05A_2=1.05\times 1.05A_1=A_1{\left(1.05\right)}^2\\ A_4=A_3+0.05A_3=A_3\left(1+0.05\right)=1.05A_3=1.05\times A_1{\left(1.05\right)}^2=A_1{\left(1.05\right)}^3\end{array}$

Since $A_1=32000$ , the formula for his salary in nth year can be given as

  $A_n=32000{\left(1.05\right)}^{n-1}$

Conclusion:

The formula for his salary in the nth year is $A_n=32000{\left(1.05\right)}^{n-1}$

b.

To determine

To list: the salary of the teacher for the first eight years.

Answer to Problem 29RE

  $\$32000,\$33600,\$35280,\$37044,\$38896.2,\$40841.01,\$42883.1,\$45027.2$

Explanation of Solution

Given information:

The teacher makes $32000 in the first year and gets a raise of 5% each year.

Calculation:

The formula for his salary in nth year is $A_n=32000{\left(1.05\right)}^{n-1}$.

So,

  $\begin{array}{l}{A}_1=32000{\left(1.05\right)}^{1-1}=32000\cdot {\left(1.05\right)}^0=32000\cdot 1=32000\\ A_2=32000{\left(1.05\right)}^{2-1}=32000\cdot {\left(1.05\right)}^1=32000\cdot 1.05=33600\\ A_3=32000{\left(1.05\right)}^{3-1}=32000\cdot {\left(1.05\right)}^2=35280\\ A_4=32000{\left(1.05\right)}^{4-1}=32000\cdot {\left(1.05\right)}^3=37044\\ A_5=32000{\left(1.05\right)}^{5-1}=32000\cdot {\left(1.05\right)}^4=38896.2\\ A_6=32000{\left(1.05\right)}^{6-1}=32000\cdot {\left(1.05\right)}^5=40841.01\\ A_7=32000{\left(1.05\right)}^{7-1}=32000\cdot {\left(1.05\right)}^6=42883.1\\ A_8=32000{\left(1.05\right)}^{8-1}=32000\cdot {\left(1.05\right)}^7=45027.2\end{array}$

Conclusion:

His salary for the first 8 years are $\$32000,\$33600,\$35280,\$37044,\$38896.2,\$40841.01,\$42883.1,\$45027.2$.