**Recursive Formula Calculator** – In maths, Recursive Formula Calculator is a type of expression which consists of an idea of one or more variables with their features, which is determined by a process or instructions that contains the function values by constantly applying a given operation many times to produce the profit.

## what is recursion?

Recursion is a process in which a function calls itself a subroutine. It allows the function to repeat many times as it calls itself in execution. It is simple and results in an oriented method because it needs minimum code to do any job or function. Recursive is a way to solve a problem by dividing it into many parts, solving them indivisibly, and combining the result. It breaks the problem into parts and solves them one by one. After that, it combines all results and provides the final solution to the problem. It uses a straightforward algorithm solution to any problem that is very hard to solve in an iterative algorithm. And to perform the above theory, we use Recursive Formula Calculator.

## Recursive Sequence Formula

In a recursive sequence, terms are defined by the previously given term. If you know the nth term of an arithmetic sequence and the common difference, d, you can find the (n+1)th term using the recursive formula **an+1=an+d**.

**Example**

Recursive_sequence(expression; first-term;upper_value; variable) Suppose, you are solving a geometric sequence using a recursive sequence calculator than you should write like Recursive_sequence(4*x,-1,3,x);

Where, variable = x, first-term = -1, upper_value = 3, expression = 4*x.

In recursive Calculators, Elementary functions are addition, multiplication, Exponentiation, and Binomial coefficients.

- In the recursive rule calculator, addition can be defined based on the counting values as (1+n)+a =1+(n+a).
- Followed by multiplication, it is defined recursively as (1+n)a = a+na.
- To determine Exponentiation in the recursive formula calculator, it will be written as a1+n = can.

**Example 1:**

Find the 9th term of the arithmetic sequence if the common difference is 77 and the 8th8th term is 5151.

**a9=a8+d**

**a9=51+7=58**

**Example 2:**

Write the first four terms of the geometric sequence whose first term is a1=3a1=3 and whose common ratio is r=2r=2.

**a1=3**

**a2=a1r=3(2)=6**

**a3=a2r=6(2)=12**

**a4=a3r=12(2)=24**

## How to Write a Recursive Formula

Let’s take an example of how we can solve this function using Recursion

- Step 1: You have a clear idea of what your function does.
- Step 2: Find out the subproblem from the function and assume your function already works on it.
- Step 3: Develop the answer to your subproblem, and use it to determine the original problem.
- % problem is solved, solve the base case to exploit the output, and you are done with it.

From the above illustration of Recursion, You are clear about its working; now we will describe how we can use this in our Recursive formula calculator.

A recursion is defined as a particular class of object that two main properties can determine:

- Base case
- Special rules to determine all other subcases

## How to Use Recursive Formula for Fibonacci Sequence

To solve any problem with a Recursive formula calculator, Please follow the below step.

- In this Recursive formula calculator, You can solve Fibonacci, arithmetic, and geometric, and select any option.
- Put input value in the field.
- Enter the value in the if-case statement. That should be used when a function or subclass is implemented and checks the eligibility and value of the function.
- Now add the function of the problem defined in the input field. Like for the Fibonacci sequence, add the function as f(n) = f(n-1)+f(n-2).
- Now, add the value of n, where n is mentioned in the function. So, it will be f(10).
- Now submit all fields, and you get the final solution to your problem.

It’s very simple to perform this calculation using our recursive calculator. You can also draw a graph of your solution.

## Arithmetic Sequence Recursive Formula

In an arithmetic sequence, the difference between successive terms is always constant. It may be positive or negative on the sign and result in an arithmetic sequence leaning towards positive or negative infinity.

**Arithmetic sequence Written like this**

an = a1 + f × (n-1) – or more generally where **a** refers to the **nth **term in the sequence

an = am + f × (n-m) – **a1** is the first term

i.e. a1, a1 + f, a1 + 2f, … – **f** is the common difference

EX: 1, 3, 5, 7, 9, 11, 13, …

### Geometric Sequence Recursive Formula

In a geometric sequence, Every next number after the first number is the multiplication of the previous number with a fixed, non-zero number.

**Geometric sequences Written like this**

an = a × rn-1 – where **a** refers to the **nth** term in the sequence

i.e. a, ar, ar2, ar3, … – **a** is the scale factor and **r** is the common ratio

EX: 1, 2, 4, 8, 16, 32, 64, 128,

### Fibonacci Sequence Recursive Formula

In the Fibonacci sequence, every number following the first two is the sum of the preceding numbers. The first two numbers always are 1-0 and 0-1, depending on starting chosen point.

**Fibonacci sequence is written as:**

an = an-1 + an-2 – where **a** refers to the **nth** term in the sequence

EX: 0, 1, 1, 2, 3, 5, 8, 13, 21, … a0 = 0; a1 = 1

## FAQ

**What Is a _{n} in Recursive Formula?**

a_{n} refers to the n^{th} term in the sequence

**What is the Recursive Formula in Math?**

A **recursive formula **Defines any term of sequence in terms of its preceding term.

**How to Find the Recursive Formula for an Arithmetic Sequence?**

- Step 1: Identify the n
^{th}term (a_{n}) of an arithmetic sequence and the common difference, d, - Step 2: Put the values in the formula, a
_{n+1}= a_{n}+ d to find the (n+1)^{th}term to find the successive terms.