Unit 6: Practical Applications
This unit explores the practical applications of infinite geometric series through real-world scenarios and examples. Understanding these applications enhances your ability to apply mathematical concepts to solve complex problems.
Introduction
Infinite geometric series have numerous applications across various fields, including finance, engineering, architecture, and more. This unit delves into a detailed worked example and explores different scenarios where infinite geometric series play a crucial role.
Worked Example: Infinite Geometric Series
Scenario:
A sequence of squares is formed by joining the midpoints of the sides of the previous square. The initial square has sides of length \(12x\) cm. The process is repeated infinitely.
Side Lengths of the Squares:
- Initial square side: \(12x\).
- Next square side (using Pythagoras): \[ \text{New side} = 6\sqrt{2}x \]
- Subsequent sides form a geometric sequence.
Areas of the Squares:
- Initial area: \(144x^2\).
- Next area: \(72x^2\).
- Subsequent areas: \(36x^2\), \(18x^2\), \(\dots\)
The areas form a geometric series with:
- First term (\(a\)): \(144x^2\).
- Common ratio (\(r\)): \(\frac{1}{2}\).
Sum of Infinite Areas:
The formula for the sum of an infinite geometric series is:
\[ S_{\infty} = \frac{a}{1 - r} \]
Substituting the values:
\[ S_{\infty} = \frac{144x^2}{1 - \frac{1}{2}} = \frac{144x^2}{\frac{1}{2}} = 288x^2 \]
Conclusion: The total area is \(288x^2\).
Practical Applications of Series
- Depreciation of Assets: Calculating the reducing balance value of equipment over time.
- Savings Plans: Determining the amount saved in a recurring deposit with fixed interest rates.
- Infinite Series in Architecture: Design applications like fractal patterns or tessellations that repeat infinitely.
Flashcards
A geometric series that continues indefinitely with no finite number of terms.
\[ S_{\infty} = \frac{a}{1 - r}, \quad \text{where } |r| < 1 \]
When the absolute value of the common ratio \(r\) is less than 1, i.e., \(|r| < 1\).
Practice Questions
- Evaluate \(\sum_{k=1}^{5} \left(\frac{3}{4}\right)^k\).
- If the first term of a geometric series is 200 and the common ratio is 0.75, find the sum to infinity.
- Determine the sum of the series \(\sum_{n=0}^{\infty} 5 \cdot \left(\frac{1}{3}\right)^n\).
- The area of each subsequent circle in a sequence is half of the previous one. If the first area is \(50\pi\), find the total area of all circles.
Quick Quiz
Summary
- Infinite geometric series are sums of infinitely many terms where each term is a constant multiple of the previous one.
- The sum to infinity converges if the absolute value of the common ratio \(r\) is less than 1.
- Understanding infinite geometric series is essential for applications in finance, engineering, architecture, and various other fields.