Chapter 9, Problem 63RE

(a)

To determine

Find the formula for the $nth$ of a geometric sequence that gives the value of the machine $t$ full years after it is purchased.

Answer to Problem 63RE

The value of the machine after n years is the nth term of the finite geometric series

  $a_n=120000{\left(0.7\right)}^{n-1}$

Explanation of Solution

Given:

Cost of machine $120,000

Rate of depreciates 30% per year

At the end of each year the depreciated value is 70% of what it was at the beginning of the year.

Let’s find the formula for value of the machine after n years

The $nth$ term of a finite geometric series is given by

  $a_n=a_1{r}^{n-1}$

Here, the initial price of the machine, that is

  $a_1=120000$

Machine's value depreciates 30% each year, which means that

Value of the machine in a particular year is 70% of that in the previous year. That is,

  $\begin{array}{l}r=70\%or\hfill \\ r=0.7\hfill \end{array}$

Thus, the value of the machine after n years is the nth term of the finite geometric series

  $a_n=120000{\left(0.7\right)}^{n-1}$

(b)

To determine

Find the depreciated value of the machine after 5 full years

Answer to Problem 63RE

Depreciated value of the machine after five years is $28,812.

Explanation of Solution

Given:

Cost of machine $120,000

Rate of depreciates 30% per year

At the end of each year the depreciated value is 70% of what it was at the beginning of the year.

To find the value of the machine after a specific period of time

Here, to find the depreciated value of the machine after 5 years

  $\begin{array}{l}{a}_1=120000\hfill \\ r=0.7\hfill \\ a_5=a_1{r}^{5-1}\hfill \\ a_5=120000{\left(0.7\right)}^4\hfill \\ a_5=28812\hfill \end{array}$

Depreciated value of the machine after five years is $28,812.