10 Practical Examples of Geometric Series in Real-World Contexts

📚 Quick Study Guide

• 🔢 A geometric series is a sum of terms where each term is multiplied by a constant ratio ($r$) from the previous term.
• 📈 The general form of a geometric series is: $a + ar + ar^2 + ar^3 + ...$ where $a$ is the first term and $r$ is the common ratio.
• 🧮 The sum of a finite geometric series with $n$ terms is given by: $S_n = \frac{a(1 - r^n)}{1 - r}$, where $r \neq 1$.
• ♾️ An infinite geometric series converges (has a finite sum) if $|r| < 1$, and its sum is: $S = \frac{a}{1 - r}$.
• 📅 Geometric sequences and series are used in finance (compound interest), physics (damped oscillations), and computer science (algorithm analysis).

Practice Quiz

1. A ball is dropped from a height of 10 meters. Each time it hits the ground, it bounces to 3/4 of its previous height. What is the total vertical distance traveled by the ball?

   1. 20 meters
   2. 50 meters
   3. 70 meters
   4. 100 meters

2. A company offers a job with a starting salary of $40,000 and promises a 3% raise each year. What will the salary be in 5 years?

   1. $44,000
   2. $46,370.91
   3. $52,000
   4. $60,000

3. What is the sum of the infinite geometric series: $1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + ...$?

   1. $\frac{1}{2}$
   2. $\frac{2}{3}$
   3. $\frac{3}{2}$
   4. 2

4. A population of bacteria doubles every hour. If you start with 100 bacteria, how many will there be after 4 hours?

   1. 400
   2. 800
   3. 1200
   4. 1600

5. You invest $1000 in an account that earns 5% interest compounded annually. What will be the value of the investment after 10 years?

   1. $1500
   2. $1628.89
   3. $1710.34
   4. $1800

6. A pendulum swings such that each swing is 90% of the length of the previous swing. If the first swing is 2 meters long, what is the total distance the pendulum swings?

   1. 10 meters
   2. 15 meters
   3. 20 meters
   4. 25 meters

7. A fractal is constructed such that each iteration adds 4 copies of the previous shape, scaled down by a factor of 1/3. If the initial shape has an area of 1, what is the area after infinite iterations?

   1. $\frac{3}{5}$
   2. $\frac{4}{5}$
   3. $\frac{5}{3}$
   4. $\frac{5}{4}$