How to Write an Explicit Formula for Any Sequence: A Step-by-Step Guide

📚 What is an Explicit Formula?

An explicit formula is a mathematical expression that allows you to directly calculate any term of a sequence using its position (usually denoted as $n$). Unlike recursive formulas, which rely on previous terms, explicit formulas provide a direct route to finding any term in the sequence. They are incredibly useful for dealing with large sequences or when you need a specific term without calculating all the preceding ones.

📜 A Brief History

The concept of sequences and series has ancient roots, dating back to early civilizations like the Babylonians and Greeks. However, the formalization of explicit formulas came later, with the development of algebraic notation and calculus. Mathematicians like Fibonacci (with his famous sequence) and others contributed to our understanding of how to represent patterns mathematically, eventually leading to the explicit formulas we use today.

📌 Key Principles for Writing Explicit Formulas

• 🔍 Identify the Pattern: Look for a consistent relationship between the term number ($n$) and the term value. Is it arithmetic (constant difference), geometric (constant ratio), or something else?
• 🔢 Arithmetic Sequences: If the sequence has a constant difference ($d$), the explicit formula is typically of the form $a_n = a_1 + (n-1)d$, where $a_1$ is the first term.
• ✖️ Geometric Sequences: If the sequence has a constant ratio ($r$), the explicit formula is typically of the form $a_n = a_1 * r^{(n-1)}$, where $a_1$ is the first term.
• 💡 Adjust and Simplify: After finding a potential formula, test it with several terms to ensure it holds true. Simplify the expression as much as possible.
• 📝 Consider Special Cases: Some sequences may require piecewise formulas or adjustments based on the term number.

🌍 Real-World Examples

Let's look at some examples to clarify how to write explicit formulas:

1. Arithmetic Sequence: Consider the sequence 3, 7, 11, 15, ... The common difference is 4. The explicit formula is $a_n = 3 + (n-1)4 = 4n - 1$.
2. Geometric Sequence: Consider the sequence 2, 6, 18, 54, ... The common ratio is 3. The explicit formula is $a_n = 2 * 3^{(n-1)}$.
3. Square Numbers: Consider the sequence 1, 4, 9, 16, ... The explicit formula is $a_n = n^2$.

🧮 Practice Problems

Write the explicit formula for the following sequences:

1. 5, 10, 15, 20, ...
2. 1, -2, 4, -8, ...
3. 2, 5, 8, 11, ...

🧪 Solutions

1. $a_n = 5n$
2. $a_n = (-2)^{(n-1)}$
3. $a_n = 3n - 1$

🔑 Conclusion

Writing explicit formulas is a powerful tool for understanding and working with sequences. By identifying patterns, applying the appropriate formula structure, and testing your results, you can master this essential mathematical skill.