Real-World Examples of Sequences in High School Math

📚 Quick Study Guide

🔢
• Arithmetic Sequence: A sequence where the difference between consecutive terms is constant. Formula: $a_n = a_1 + (n-1)d$, where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.
📈
• Geometric Sequence: A sequence where each term is multiplied by a constant to get the next term. Formula: $a_n = a_1 * r^(n-1)$, where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the term number, and $r$ is the common ratio.
💻
• Fibonacci Sequence: A sequence where each term is the sum of the two preceding terms. Starts with 0 and 1. Example: 0, 1, 1, 2, 3, 5, 8...
➕
• Recursive Formula: Defines a term based on the previous term(s).
💡
• Real-World Examples: Include compound interest, population growth, and patterns in nature.

✍️ Practice Quiz

1. Question 1: A stack of boxes is arranged such that the bottom row has 15 boxes, the next row has 13, and so on, with each row having 2 fewer boxes than the row below it. This is an example of what kind of sequence?
   1. Arithmetic
   2. Geometric
   3. Fibonacci
   4. Random
2. Question 2: A population of bacteria doubles every hour. If you start with 10 bacteria, what type of sequence models the population growth?
   1. Arithmetic
   2. Geometric
   3. Fibonacci
   4. None of the above
3. Question 3: What is the next number in the following sequence: 1, 1, 2, 3, 5, 8, ...?
   1. 10
   2. 12
   3. 13
   4. 15
4. Question 4: You deposit $100 into a savings account that earns 5% interest compounded annually. The amounts in your account each year form what kind of sequence?
   1. Arithmetic
   2. Geometric
   3. Fibonacci
   4. None of the above
5. Question 5: A staircase is designed so that each step is 7 inches high. What kind of sequence represents the height you reach after each step?
   1. Arithmetic
   2. Geometric
   3. Fibonacci
   4. Random
6. Question 6: Which of the following real-world scenarios best represents a Fibonacci sequence?
   1. The height of a bouncing ball after each bounce.
   2. The arrangement of seeds in a sunflower.
   3. The decay of a radioactive substance.
   4. The number of cars passing a point each hour.
7. Question 7: A tree grows 1 foot each year. If it was initially 4 feet tall, what type of sequence represents its height each year?
   1. Arithmetic
   2. Geometric
   3. Fibonacci
   4. Exponential