Recursive Formula Calculator (with Examples)

A recursive formula helps calculate a sequence where each number depends on the one before it.
Use our calculator to easily generate terms in a recursive sequence and see how each step unfolds. 📈

🙋‍♀️ What Is a Recursive Formula?

A recursive formula defines a term in a sequence using previous terms.
Unlike an explicit formula, which calculates a term directly, a recursive formula uses a pattern like:

a₁ = 3  
aₙ = aₙ₋₁ + 2

This means:

• The first term is 3
• Each subsequent term is 2 more than the previous one

This approach is commonly used in:

• Math sequences
• Computer algorithms
• Fibonacci numbers
• Modeling growth and patterns over time

🛠️ How to Use the Recursive Formula Calculator

1. Enter the first term
2. Define your recursive rule (e.g., a + 3, a * 2, a * a + 1)
3. Choose the number of terms you want
4. See the full sequence instantly!

🧮 Recursive Formula Format

The general format is:

a₁ = starting value  
aₙ = function of aₙ₋₁

You can define the rule with any mathematical operation:

• Addition: aₙ = aₙ₋₁ + 5
• Multiplication: aₙ = aₙ₋₁ × 2
• Custom functions: aₙ = (aₙ₋₁ + 1)²

📈 Calculation Example

Let’s say:

a₁ = 4  
aₙ = aₙ₋₁ × 3

The calculator will produce:

• a₁ = 4
• a₂ = 12
• a₃ = 36
• a₄ = 108
• a₅ = 324

You’ll see how each term is derived from the previous one.

🎯 Why Use This Calculator?

• ✅ Quickly generate long recursive sequences
• ✅ Avoid manual repetition and math errors
• ✅ Great for students, teachers, coders, and researchers
• ✅ Understand how recursion builds a pattern step-by-step

✨ Final Tip

Recursive formulas are a fundamental tool in both mathematics and programming. By understanding them — and with the use of this calculator — you’ll save time and gain deeper insight into patterns and logic.