2. Case Study Question

An engineering company is monitoring the value of an advanced machine over several years. Due to technological depreciation, the machine loses a fixed percentage of its value at the end of every year. The management team wants to study the pattern formed by the machine's value and use it to predict its value in future years.
The company purchases the machine for \(₹1562500\). Its value decreases by \(20\%\) at the end of every year. After several years, the machine is valued at at \(₹640000\).

 

1. What percentage of its previous year's value does the machine retain each year?

 

2. Show that the values of the machine form a geometric progression and determine its common ratio.

 

3. Write the explicit formula for the \(n^{\text{th}}\) term of the geometric progression.

 

• $t_n=1562500{\left(\frac{4}{5}\right)}^n$
• $t_n=1562500{\left(\frac{5}{4}\right)}^{n+1}$
• $t_n=1562500{\left(\frac{4}{5}\right)}^{n-1}$
• $t_n=1562500{\left(\frac{5}{4}\right)}^{n-1}$

4. After how many years will the value of the machine become \(₹640000\)?