Chapter 12, Problem 15CT

To determine

To calculate: An amount a new car will be worth after 10 years, if it is sold for $\$31,000$ and if the vehicle loses $15\%$ of its value each year.

Answer to Problem 15CT

Solution:

A new car will be worth $\$6,103.11$ after 10 years.

Explanation of Solution

Given information:

A new car is sold for $\$31,000$ and it loses $15\%$ of its value each year.

Formula used:

$n^{th}$ term of the geometric sequence $\left\{a_n\right\}$ with $a_1$ as first term and $r$ as common ratio is

$a_n=a_1\cdot {(r)}^{n-1}$.

Calculation:

As new car loses $15\%$ of its value each year, the value for the next year is $85\%$.

Thus yearly values of the car form a geometric series.

$a_1=31,000,r=1-0.15=0.85$.

Thus nth term,

$a_n=31,000\cdot {(0.85)}^{n-1}$.

The nth term in the formula gives the value of the car in the beginning of the year.

For the value of the car after 10 years, $n=11$.

Thus,

$a_{11}=31000\cdot {(0.85)}^{11-1}\approx 6103.11$

Therefore, a new car will be worth $\$6,103.11$ after 10 years.