1. What Is a Recursive Sequence?

A recursive sequence is a sequence where each term is defined as a function of the preceding terms. In this example, we use the formula aₙ = 2 × aₙ₋₁, where each term is twice the previous term.

2. How Does the Calculator Work?

The calculator uses the recursive formula:

Where:

• \( a_n \) — Current term in the sequence
• \( a_{n-1} \) — Previous term in the sequence
• 2 — The multiplication factor

Explanation: Starting from an initial value, each subsequent term is calculated by multiplying the previous term by 2.

3. Importance of Recursive Sequences

Details: Recursive sequences are fundamental in mathematics, computer science, and modeling real-world phenomena like population growth, financial calculations, and fractal patterns.

4. Using the Calculator

Tips: Enter an initial value and the number of terms you want to generate. The calculator will display the sequence starting from the initial term.

5. Frequently Asked Questions (FAQ)

Q1: What types of recursive sequences exist?
A: Besides geometric sequences like this one, there are arithmetic sequences, Fibonacci sequences, and many other recursive patterns with different relationships between terms.

Q2: Can I use different multiplication factors?
A: Yes, this calculator specifically uses a factor of 2, but recursive sequences can use any constant factor or even variable factors.

Q3: What are practical applications of this sequence?
A: This specific sequence (aₙ = 2 × aₙ₋₁) models exponential growth seen in bacterial division, compound interest, and computer algorithms.

Q4: How does this differ from an explicit formula?
A: A recursive formula defines terms relative to previous terms, while an explicit formula defines each term independently based on its position.

Q5: What's the explicit formula for this sequence?
A: For aₙ = 2 × aₙ₋₁ with initial value a₁, the explicit formula is aₙ = a₁ × 2ⁿ⁻¹.