Chapter 12.2, Problem 67AYU

Salary If you take a job with a starting salary of $\$35,000$ per year and a guaranteed raise of $\$1400$ per year, how many years will it be before your aggregate salary is $\$280,000$?

[Hint: Remember that your aggregate salary after 2 years is $\$35,000+\left(\$35,000+\$1400\right)$.]

To determine

To find: If you take a job with a starting salary of $\$35,000$ per year and a guaranteed raise of $\$1400$ per year, how many years will it be before your aggregate salary is $\$280,000$? [Hint: Remember that your aggregate salary after 2 years is $\$35,000+\left(\$35,000+\$1400\right)$.]

Answer to Problem 67AYU

8 Years.

Explanation of Solution

Given:

Starting salary $=\$35,000$.

Annual increase $=\$1,400$.

Formula used:

Sum of the first $n$ terms of an arithmetic sequence

$S_n=\frac{n}{2}\left[2a_1+\left(n-1\right)d\right]$

Calculation:

From the given data, we get $a_1=35000,S_n=280000,d=1400$.

From the above formula,

$280000=\frac{n}{2}\left[2\times 35000+\left(n-1\right)\times 1400\right]$

Reducing the above equation, we get

$n^2+49n-400=0$

Solving the above quadratic equation,

$n=\frac{-49\pm \sqrt{2401+1600}}{2}$

$n=7.13$   (Since number of years cannot be negative)

The number of years before the aggregate salary is $\$280,000=8\text{ years}$.