Question

An electronic games manufacturer producing a new product estimates the annual sales to be 8000 units. Each year, \(10 \%\) of the units that have been sold will become inoperative. So, 8000 units will be in use after 1 year, \([8000+0.9(8000)]\) units will be in use after 2 years, and so on. How many units will be in use after \(n\) years?

Short answer

The number of units in use after n years is given by the formula \(S(n) = 8000 * (1 - 0.9^n) / 0.1\)

Step-by-step solution

Understand the problem

Create an infinite Geometric Progression (GP) series where the first term (a) is 8000 units (number of units produced in the first year) and the common ratio (r) is 0.9 (90% of units continue to work next year). In other words, series would look like: 8000, 8000 + 0.9 * 8000, 8000 + 0.9 * 8000 + 0.9^2 * 8000,... and so on.

Formulate the series

Next, denote the total number of units still operative after n years as S(n). Then write down the formula for the sum of the first n terms of a GP - \(S(n) = a * (1 - r^n) / (1 - r)\). Replace \(a\) with 8000 and \(r\) with 0.9.

Simplify the equation

Substitute \(a = 8000\), and \(r = 0.9\) into the formula, we could obtain the number of units that will be in use after n years, which gives us: \(S(n) = 8000 * (1 - 0.9^n) / (1 - 0.9)\).

Conclusion

So, it can be concluded that the number of units in use after n years can be found by using the equation \(S(n) = 8000 * (1 - 0.9^n) / 0.1\). This will give the number of units from each year that are still operating, summed up.