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Recursive Formula Calculator

Enter the required values into the recursive formula calculator to find the next term in a sequence.

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Given the recursive formula an = 3 an−1 + 4, where a1 =2, find the value of a2, a3 and a4

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Recursive Formula Calculator

Our Recursive Formula Calculator helps you compute terms in a sequence using previous terms and a defined rule. It provides step-by-step solutions, making it easier to understand how each term is generated. This tool is perfect for algebra and pre-calculus students learning about sequences and series. You can calculate specific terms (like a₁ to a₁₀) or generate a list of terms up to a desired point.

What is a Recursive Formula?

A recursive formula is a mathematical rule that defines each term of a sequence based on the preceding term(s). Instead of providing a direct formula for the nth term, it shows how to compute the next term from the previous one.

General Recursive Formula:

a(n) = a(n−1) + d

Where:

a(n) = Nth term in the sequence
a(n−1) = Previous term
d = Common difference between terms (for arithmetic sequences)

How to Use the Recursive Formula Calculator

Enter the recursive formula: e.g., aₙ = 2aₙ₋₁ + 3
Input the initial term: Provide the first term of the sequence (e.g., a₁ = 1)
Specify which terms to calculate: e.g., a₂, a₃, a₄, …
Click Calculate: The tool will compute terms step-by-step using the recursive rule.

How to Solve Recursive Sequences

Steps to compute subsequent sequence values:

Start with the initial term a(1)
Identify the rule or common difference d (if arithmetic)
Apply the recursive formula: a(n) = a(n−1) + d
Calculate the nth term a(n)

Example

Given:

Recursive Rule: a(n) = a(n−1) − 4
Initial Term: a(1) = 10
Compute the first 6 terms

Solution:

a(2) = a(1) − 4 = 10 − 4 = 6

a(3) = a(2) − 4 = 6 − 4 = 2

a(4) = a(3) − 4 = 2 − 4 = −2

a(5) = a(4) − 4 = −2 − 4 = −6

a(6) = a(5) − 4 = −6 − 4 = −10

FAQs

What is the recursive formula for 2, 4, 6, 8, 10?

This arithmetic sequence can be written recursively as: a(n) = a(n−1) + 2, with a₁ = 2.

Can you give a recursive equation example?

Example: a(n) = a(n−1) + 3, with a(1) = 2

Start: a(1) = 2
Each term = previous term + 3

Sequence: 2, 5, 8, 11, 14, …

Why do I need initial conditions for a recursive formula?

Initial conditions provide the starting point. Without them, the recursive formula cannot generate any terms.

Can a recursive formula define all sequences?

Yes, any sequence can be defined recursively once the relationship between successive terms is known.

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