Problem 91 - Essential Calculus

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?

Step-by-Step Solution

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 1: 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.

Step 2: 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.

Step 3: 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: \(S(n) = 8000 * (1 - 0.9^n) / (1 - 0.9)\).

Step 4: 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.

Key Concepts

Formula Derivation

A Geometric Progression (GP) is a sequence of numbers in which each term after the first is found by multiplying the previous term by a constant known as the 'common ratio'. The GP formula for the sum of the first 'n' terms can be derived using a simple manipulation of algebraic expressions.
To derive the formula, let's first represent a GP:

• The first term is denoted by 'a'.
• The common ratio, 'r', is the fixed multiplier between terms.

Given these variables, the series would be:

• a
• a \( \times \) r
• a \( \times \) r^2
• ...
• a \( \times \) r^{n-1}

To find the sum of the series up to 'n' terms, denoted as \( S(n) \), we use the formula: \[ S(n) = a\frac{(1 - r^n)}{1 - r} \] This formula is derived by multiplying and manipulating the terms of the geometric progression to observe a consistent cancellation pattern that simplifies to the above expression.
In simpler terms, it is all about arranging terms and using common cancellations to get to a clean expression for the sum.

Series Summation

In the given exercise, we need to determine how many of the units will continue to be in operation after multiple years. To solve this, we model the problem using a geometric series where we know the first term and the common ratio.
Here, the first term, 'a', is 8000, which is the number of units produced initially. The common ratio, 'r', is 0.9 because each year, 90% of the units will still be functioning.

• The sequence of in-use units over the years can be represented as: 8000, 8000 \(+\) 0.9 \( \times \) 8000, 8000 \(+\) 0.9 \( \times \) 8000 \(+\) 0.9^2 \( \times \) 8000, and so on.

To find out how many units are still in use after 'n' years, we calculate the sum of the series using the formula: \[ S(n) = 8000\left(\frac{1 - 0.9^n}{1 - 0.9}\right) \] This formula allows us to add up all the remaining units that still function each year up to the 'n' year.

Mathematical Problem Solving

Practical applications of geometric progression involve translating real-world problems, such as this series of non-operating units, into mathematical expressions that can be solved to yield useful results.
When given an annual sales problem like this one, the initial step is to understand the continuous nature of the operations or sales; here, it's the decrease in operational units over years.

• We begin by identifying the first term, which is often the initial condition—in this case, the number of units initially sold, 8000.
• Next, recognize the systematic decrease or operation which will form the common ratio—in this scenario, it's 0.9, representing 90% continuation.

By framing the problem into a known mathematical structure such as a GP, we use formula manipulation to make predictions or solve for variables like total operating units after 'n' years.
This approach highlights the beauty and power of mathematics in providing solutions to operational and sales forecasting issues in business contexts.