Real-World Examples of Geometric Sequences & Their Applications

📚 Quick Study Guide

• 🔢 A geometric sequence is a sequence where each term is multiplied by a constant value to get the next term. This constant is called the common ratio ($r$).
• 📈 The general form of a geometric sequence is: $a, ar, ar^2, ar^3, ...$ where $a$ is the first term.
• 💰 Compound interest is a classic example: The amount of money grows geometrically each year. The formula is: $A = P(1 + i)^n$, where $A$ is the final amount, $P$ is the principal, $i$ is the interest rate, and $n$ is the number of years.
• 🦠 Population growth (under ideal conditions) can be modeled geometrically.
• ☢️ Radioactive decay also follows a geometric pattern. The amount of radioactive material decreases geometrically over time.
• 📐 Fractals often exhibit geometric sequences in their construction.

Practice Quiz

1. A ball is dropped from a height of 16 feet. Each time it bounces, it rebounds to one-half of its previous height. What is the height of the ball after the 3rd bounce?

   1. 2 feet
   2. 4 feet
   3. 1 foot
   4. 8 feet

2. A bacteria culture doubles in size every hour. If the initial population is 1000, what will the population be after 4 hours?

   1. 4000
   2. 8000
   3. 16000
   4. 32000

3. What is the 7th term of the geometric sequence: 2, 6, 18, ...?

   1. 1458
   2. 486
   3. 4374
   4. 13122

4. An antique car appreciates in value by 8% each year. If its initial value is $25,000, what will its approximate value be after 5 years?

   1. $35,000
   2. $36,733
   3. $26,000
   4. $30,000

5. A radioactive substance decays such that half of it is gone every 10 years. If you start with 100 grams, how much will be left after 30 years?

   1. 50 grams
   2. 25 grams
   3. 12.5 grams
   4. 6.25 grams

6. Find the sum of the first 5 terms of the geometric sequence: 3, 6, 12, ...

   1. 93
   2. 48
   3. 186
   4. 36

7. A company offers a job with a starting salary of $40,000 and a 3% annual raise. What will the salary be in the 10th year?

   1. $53,756.73
   2. $51,621.87
   3. $41,200.00
   4. $60,000.00