Chapter 8, Problem 13CR

To determine

Find the rule for the $n\text{th}$ year of the given salary and total earning in the given years.

Answer to Problem 13CR

Thetotal earnings in $6$ years are $\$2,44,500$ .

Explanation of Solution

Given:

The starting salary is $\$37,000$ and annual raise is $\$1500$ for the next six years.

Calculation:

The general rule for the $n\text{th}$ term is $a_n=a_1+\left(n-1\right)d$ .

Where $a_1\to$ first term, $d\to$ common difference, $n\to$ number of terms.

In the given data

  $\begin{array}{l}{a}_1=37,000\\ d=1500\end{array}$

The $n\text{th}$ term $a_n=37000+\left(n-1\right)1500$

First salary $a_1=37,000$

Last $\left(6\text{th}\right)$ salary

  $\begin{array}{l}{a}_6=37000+\left(6-1\right)1500\\ =37000+7500\\ =44500\end{array}$

The total earnings in $6$ years

  $\begin{array}{l}s=\left(\frac{a_1+a_n }{2}\right)n\\ =\left(\frac{37000+44500 }{2}\right)6\\ =81500\cdot 3\\ =\$2,44,500\end{array}$

Hence the total earnings in $6$ yearsare $\$2,44,500$ .