1. What is a Recursive Formula?

A recursive formula defines each term of a sequence using the preceding term(s). It provides a step-by-step method for generating the terms of a sequence rather than a direct formula for the nth term.

2. How Does the Calculator Work?

The calculator uses recursive formulas for arithmetic and geometric sequences:

Where:

• \( a_n \) — The next term in the sequence
• \( a_{n-1} \) — The previous term in the sequence
• \( d \) — Common difference (for arithmetic sequences)
• \( r \) — Common ratio (for geometric sequences)

3. Types of Recursive Formulas

Arithmetic Sequences: Each term is obtained by adding a constant value (common difference) to the previous term.

Geometric Sequences: Each term is obtained by multiplying the previous term by a constant value (common ratio).

4. Using the Calculator

Steps: Select the sequence type, enter the previous term (a_{n-1}), and enter either the common difference (for arithmetic) or common ratio (for geometric). The calculator will compute the next term (a_n).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between recursive and explicit formulas?
A: Recursive formulas define terms relative to previous terms, while explicit formulas define the nth term directly in terms of n.

Q2: Can I use this for sequences with more complex patterns?
A: This calculator handles only simple arithmetic and geometric sequences. More complex recursive sequences may require specialized tools.

Q3: What if the common ratio is zero in a geometric sequence?
A: If r = 0, all subsequent terms become zero after the first non-zero term.

Q4: How do I find the common difference or ratio from given terms?
A: For arithmetic: d = a_n - a_{n-1}. For geometric: r = a_n / a_{n-1}.

Q5: Can recursive formulas be used for decreasing sequences?
A: Yes, arithmetic sequences decrease when d is negative, and geometric sequences decrease when 0 < r < 1.