Understanding Exponential Growth and Decay

Exponential growth and decay are really cool topics. They are closely linked to something called geometric sequences. Let’s break it down in a simple way:

1. Key Ideas:

   • Exponential Growth: Imagine your money in a bank account that doubles over time. That’s how growth works!
   • Exponential Decay: This is when something gets smaller. For example, think about how radioactive materials break down over time or how a car loses value.

2. Geometric Sequences: These sequences are a way to show both growth and decay. In a geometric sequence, each part can be figured out using a special formula:

   • ( a_n = a_1 \cdot r^{(n-1)} ) Here, ( a_1 ) is the first number in the sequence, and ( r ) is the number you multiply by.

3. Adding the Terms Up: If you want to find out the total of the first few numbers in a geometric sequence, you can use this formula:

   • ( S_n = a_1 \frac{1 - r^n}{1 - r} ) (as long as ( r ) is not 1)

So, both growth and decay are like watching geometric sequences do their thing!