Formula for the nth term of a geometric sequence

📚 What is a Geometric Sequence?

A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a constant value called the common ratio. Think of it like repeatedly scaling a number up or down.

📜 History and Background

Geometric sequences have been studied for centuries, appearing in ancient mathematical texts. Their properties are fundamental to understanding exponential growth and decay, seen everywhere from compound interest to radioactive decay.

🔑 Key Principles: The Formula

The formula for finding the $n$th term ($a_n$) of a geometric sequence is:

$a_n = a_1 * r^{(n-1)}$

Where:

• 🍎 $a_n$ represents the $n$th term in the sequence.
• 🔢 $a_1$ is the first term of the sequence.
• ⚖️ $r$ is the common ratio (the constant value multiplied to get the next term).
• 📍 $n$ is the position of the term you want to find.

🧮 How to Use the Formula

Here’s how to apply the formula in practice:

1. 🔍 Identify the first term ($a_1$) of the sequence.
2. 📐 Determine the common ratio ($r$) by dividing any term by its preceding term.
3. ✍️ Plug the values of $a_1$, $r$, and $n$ (the term you want to find) into the formula.
4. ✅ Calculate $a_n$ to find the $n$th term.

🌍 Real-World Examples

Geometric sequences appear in various real-world scenarios:

• 💰 Compound Interest: The amount of money in a bank account grows geometrically each year if interest is compounded annually.
• ☢️ Radioactive Decay: The amount of a radioactive substance decreases geometrically over time.
• 🧬 Population Growth: Under ideal conditions, populations can grow geometrically.

➗ Example Problems with Solutions

Problem 1: Find the 7th term of the geometric sequence 2, 6, 18, ...

Solution:

• 🔍 $a_1 = 2$
• 📐 $r = 6 / 2 = 3$

Using the formula:

$a_7 = 2 * 3^{(7-1)} = 2 * 3^6 = 2 * 729 = 1458$

Therefore, the 7th term is 1458.

Problem 2: The first term of a geometric sequence is 5, and the common ratio is -2. Find the 5th term.

Solution:

• 🔍 $a_1 = 5$
• 📐 $r = -2$

Using the formula:

$a_5 = 5 * (-2)^{(5-1)} = 5 * (-2)^4 = 5 * 16 = 80$

Therefore, the 5th term is 80.

✍️ Practice Quiz

Test your understanding with these practice questions:

1. ❓ Find the 6th term of the sequence: 1, 4, 16, ...
2. ❓ The first term is 3 and the common ratio is 2. What is the 4th term?
3. ❓ Determine the 8th term of the sequence: 5, -10, 20, ...

✅ Conclusion

Understanding the formula for the $n$th term of a geometric sequence is crucial for solving various math problems and understanding real-world phenomena. With practice, you can master this concept! 🎉