Deriving Recursive Formulas for Geometric Sequences: A Complete Process

📚 Understanding Geometric Sequences

A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant. This constant is called the common ratio, often denoted as $r$.

📜 History and Background

Geometric sequences have been studied for centuries, appearing in various mathematical and practical contexts. Early mathematicians recognized the importance of understanding patterns of growth and decay, leading to the formalization of geometric sequences and their properties.

🔑 Key Principles of Recursive Formulas

Recursive formulas define a term in a sequence based on the preceding term(s). For a geometric sequence, the recursive formula consists of two parts:

• 🔢 The first term: Denoted as $a_1$.
• 🔄 The recursive step: Defines how to find $a_n$ using $a_{n-1}$.

📝 Deriving the Recursive Formula

Here’s how to derive the recursive formula for a geometric sequence:

1. 🔎 Identify the first term, $a_1$.
2. 📈 Determine the common ratio, $r$. This is the value you multiply each term by to get the next term. You can find $r$ by dividing any term by its preceding term: $r = \frac{a_n}{a_{n-1}}$.
3. ✍️ Write the recursive formula:
   • The first term: $a_1 = \text{value}$
   • The recursive step: $a_n = r \cdot a_{n-1}$ for $n > 1$

🧮 Example 1: Simple Geometric Sequence

Consider the geometric sequence: 3, 6, 12, 24, ...

• 🔎 First term: $a_1 = 3$
• 📈 Common ratio: $r = \frac{6}{3} = 2$
• ✍️ Recursive formula:
  • $a_1 = 3$
  • $a_n = 2 \cdot a_{n-1}$ for $n > 1$

🧪 Example 2: Geometric Sequence with Fractions

Consider the geometric sequence: 4, 2, 1, 0.5, ...

• 🔎 First term: $a_1 = 4$
• 📈 Common ratio: $r = \frac{2}{4} = 0.5 = \frac{1}{2}$
• ✍️ Recursive formula:
  • $a_1 = 4$
  • $a_n = \frac{1}{2} \cdot a_{n-1}$ for $n > 1$

💡 Example 3: Geometric Sequence with Negative Ratio

Consider the geometric sequence: 5, -15, 45, -135, ...

• 🔎 First term: $a_1 = 5$
• 📈 Common ratio: $r = \frac{-15}{5} = -3$
• ✍️ Recursive formula:
  • $a_1 = 5$
  • $a_n = -3 \cdot a_{n-1}$ for $n > 1$

✍️ Conclusion

Deriving recursive formulas for geometric sequences involves identifying the first term and the common ratio. By understanding these components, you can easily define any term in the sequence based on its predecessor. This approach provides a powerful tool for analyzing and understanding geometric sequences.