Recursive Formula Sequence Calculator
This calculator helps you generate terms of a sequence defined by a recursive formula. A recursive formula defines each term based on one or more preceding terms.
Use
prev for the value of the previous term (an-1) and n for the current term's index (starting from 2 for a₂).
Example: prev * 2 for a geometric sequence, prev + 5 for an arithmetic sequence, prev + n for a sequence like an = an-1 + n.
Calculated Sequence:
Enter your values and click "Calculate Sequence" to see the terms.
Understanding Recursive Formulas
A recursive formula, also known as a recurrence relation, is a way to define a sequence where each term is expressed in relation to one or more preceding terms. Unlike explicit formulas that define a term directly based on its index, recursive formulas require you to know the previous terms to find the current one.
Key Components of a Recursive Formula:
- Initial Term(s): You must be given one or more starting terms (e.g., a₁, a₂, etc.) to begin the sequence. Without these, you cannot calculate any subsequent terms.
- Recursive Rule: This is the formula that defines how to calculate the n-th term (an) using previous terms (like an-1, an-2, etc.) and sometimes the term's index (n) itself.
Common Examples of Recursive Formulas:
-
Arithmetic Sequence: Each term is found by adding a constant difference to the
previous term.
Example: a₁ = 3, an = an-1 + 4.
Sequence: 3, 7, 11, 15, …
In the calculator: Initial Term = 3, Formula =prev + 4 -
Geometric Sequence: Each term is found by multiplying the previous term by a constant ratio.
Example: a₁ = 2, an = 2 * an-1.
Sequence: 2, 4, 8, 16, …
In the calculator: Initial Term = 2, Formula =prev * 2 -
Fibonacci Sequence: Each term is the sum of the two preceding ones (requires two initial terms).
Example: F₁ = 0, F₂ = 1, Fn = Fn-1 + Fn-2.
Note: This calculator simplifies to formulas depending only onprev(an-1) andn. For Fibonacci, you'd typically need a more complex calculator or manual iteration. -
Sequences with Index 'n': Sometimes the rule depends on the term's position.
Example: a₁ = 1, an = an-1 + n.
Sequence: 1, (1+2)=3, (3+3)=6, (6+4)=10, …
In the calculator: Initial Term = 1, Formula =prev + n
How to Use the Recursive Formula Sequence Calculator:
- Initial Term (a₁): Enter the starting value of your sequence. This is the first term.
-
Recursive Formula: Input the rule for generating subsequent terms.
- Use
prevto represent the value of the immediately preceding term (an-1). - Use
nto represent the index of the term you are currently calculating (e.g., for a₂, n=2; for a₃, n=3). - You can use standard mathematical operators:
+,-,*,/,**(for exponentiation),Math.pow(),Math.sqrt(), etc.
- Use
- Number of Terms to Generate: Specify how many terms of the sequence you want the calculator to display, including the initial term.
- Click "Calculate Sequence" to see the generated terms.
Practical Applications:
Recursive formulas are fundamental in various fields:
- Computer Science: Used in algorithms (e.g., factorial, quicksort), data structures (trees, linked lists), and dynamic programming.
- Mathematics: Defining sequences, series, fractals, and in number theory.
- Finance: Modeling compound interest, loan amortization, and investment growth over time.
- Biology: Describing population growth, spread of diseases, or genetic inheritance patterns.
- Physics: Modeling iterative processes or systems that evolve based on their previous states.
This calculator provides a simple way to explore and understand how different recursive rules generate unique sequences, offering insights into their behavior and applications.