Question

shantay buys equipment for her thriving business that sells clear sunscreen for deep skin tones. the value of the equipment will decrease over time by the same percent each year. the table shows the amount the equipment will decrease in value each year for the first 4 years. use the formula for the sum of a finite geometric series, $s_n$, given below the table, to determine the total amount of depreciation over the first 7 years. round your answer to the nearest dollar. equipment depreciation per year | year | 1 | 2 | 3 | 4 | | --- | --- | --- | --- | --- | | decrease in equipment value ($) | 250 | 200 | 160 | 128 | $s_n = \frac{a(1 - r^n)}{1 - r}$ where $a$ is the first term of the sequence containing $n$ terms and $r$ is the common ratio type the number in the box. rounded to the nearest dollar, the equipment will depreciate a total of $ over the first 7 years.

Explanation:

Step1: Determine the common ratio \(r\)

Given \(a = 250\) (first - term), for \(n = 2\), \(a_2=200\).
Since \(a_n=a\times r^{n - 1}\), for \(n = 2\), \(a_2=a\times r\).
\(r=\frac{a_2}{a}=\frac{200}{250}=0.8\)

Step2: Use the sum formula for a finite geometric series \(S_n=\frac{a(1 - r^n)}{1 - r}\)

Here \(a = 250\), \(r = 0.8\), \(n = 7\)
\(S_7=\frac{250\times(1-(0.8)^7)}{1 - 0.8}\)

First, calculate \((0.8)^7=0.8\times0.8\times0.8\times0.8\times0.8\times0.8\times0.8 = 0.2097152\)

Then \(1-(0.8)^7=1 - 0.2097152 = 0.7902848\)

\(S_7=\frac{250\times0.7902848}{0.2}\)

\(250\times0.7902848 = 197.5712\)

\(S_7=\frac{197.5712}{0.2}=987.856\)

Answer:

\(988\)