# Recursive Sequence Calculator + Online Solver With Free Steps

The Recursive Sequence Calculator is used to compute the closed form of a recursive relation.

A recursive relation contains both the previous term f(n-1) and the later term f(n) of a particular sequence. It is an equation in which the value of the later term depends upon the previous term.

A recursive relation is used to determine a sequence by placing the first term in the equation.

In a recursive relation, it is necessary to specify the first term to establish a recursive sequence.

For example, the Fibonocci sequence is a recursive sequence given as:

$0,1,1,2,3,5,8,13…$

In the Fibonocci sequence, the first two terms are specified as follows:

$f(0) = 0$

$f(1) = 1$

In the Fibonocci sequence, the later term $f(n)$ depends upon the sum of the previous terms f(n-1) and f(n-2). It can be written as a recursive relation as follows:

$f(n) = f(n-1) + f(n-2)$

The term $f(n)$ represents the current term and $f(n-1)$ and $f(n-2)$ represent the previous two terms of the Fibonocci sequence.

The calculator computes the closed-form solution of the recursive equation. The closed-form solution does not depend upon the previous terms. It does not contains the terms such as $f(n-1)$ and $f(n-2)$.

For example, the equation $f(n) = 4n^{2} + 2n$ is a closed form solution as it only contains the current term $f(n)$. The equation is a function of $f(n)$ in terms of the variable $n$.

## What Is a Recursive Sequence Calculator?

The Recursive Sequence Calculator is an online tool that calculates the closed-form solution or the Recurrence equation solution by taking a recursive relation and the first term $f(1)$ as input.

The closed-form solution is a function of $n$ which is obtained from the recursive relation which is a function of the previous terms $f(n-1)$.

The Recurrence Equation Solution is calculated by solving for the first three or four terms of the recursive relation. The first term $f(1)$ specified is placed in the recursive relation and is not simplified to see a pattern in the first three or four terms.

For example, given the recursive relation:

$f(n) = f(n-1) + 3$

With the first term specified as:

$f(1) = 2$

The Recurrence Equation Solution is calculated by observing the pattern in the first four terms. The second term is calculated by placing the first term $f(1)$ in the recursive relation given above as follows:

$f(2) = f(1) + 3 = 2 + 3$

$f(2) = 5$

The third term is calculated by placing the term $f(2)$ in the recursive relation.

$f(3) = f(2) + 3 = (2 + 3) + 3$

$f(3) = 8$

Similarly, the fourth term $f(4)$ is calculated by placing the third term in the recursive relation.

$f(4) = f(3) + 3 = [(2 + 3) + 3] + 3$

$f(4) = 11$

Notice the pattern in the three equations given below:

$f(2) = 2 + 3 = 2 +3(1)$

$f(3) = (2 + 3) + 3 = 2 + 3(2)$

$f(4) = [(2 + 3) + 3] + 3 = 2 + 3(3)$

The above similar pattern in the equations formulates the closed-form solution as follows:

$f(n) = 2 + 3(n \ – \ 1)$

In this way, the Recursive Sequence Calculator computes the closed-form solution of a recursive relation given the first term. The Calculator observes the pattern in the first four terms and outputs the Recurrence Equation Solution.

## How To Use the Recursive Sequence Calculator

You can use the Recursive Sequence Calculator by following the steps given below.

The calculator can be easily used to calculate the closed-form solution from a recursive relation.

### Step 1

The user must first enter the recursive relation in the input window of the calculator. It should be entered in the block against the recursive relation function $f(n)$.

The recursive relation must contain a previous term $f(n-1)$ in the equation. The calculator sets the default recursive relation as follows:

$f(n) = 2 f(n \ – \ 1) + 1$

Where $f(n)$ is the current term and $f(n-1)$ is the previous term of a recursive sequence.

It should be noted that the user must enter the recursive relation in terms of $f$ as the calculator by default shows $f(n)$ in the input tab.

### Step 2

After entering the recursive relation, the user must then enter the first term in the block against the title $f(1)$ in the calculator’s input window. The first term is essential in calculating the recurrence equation solution of the recursive relation.

The calculator sets the first term by default as follows:

$f(1) = 1$

The term $f(1)$ represents the first term of a recursive sequence. The sequence can be written as:

$f(1),f(2),f(3),f(4),…$

### Step 3

The user must now press the “Submit” button after entering the recursive relation and the first term in the input window of the calculator.

If any input information is missing, the calculator shows in another window “Not a valid input; please try again”.

### Output

The calculator computes the closed-form solution for the particular recursive relation and shows the output in the following two windows.

#### Input

The Input window shows the input interpretation of the calculator. It shows the recursive equation $f(n)$ and the first term $f(n)$ that the user has entered.

For the default example, the calculator shows the recursive relation and the first term of the sequence as follows:

$f(n) = 2 f(n – 1) + 1$

$f(1) = 1$

From this window, the user can verify the recursive relation and the first term for which the closed-form solution is required.

#### Recurrence Equation Solution

The Recurrence Equation Solution is the closed-form solution of the recursive relation. This window shows the equation which is independent of the previous terms of a sequence. It only depends upon the current term $f(n)$.

For the default example, the calculator computes the values of the second, third and fourth terms as follows:

$f(2) = 2 f(1) + 1 = 2(1) + 1$

$f(2) = 3$

$f(3) = 2 f(2) + 1 = 2(3) + 1$

$f(3) = 7$

$f(4) = 2 f(3) + 1 = 2(7) + 1$

$f(4) = 15$

Notice the similar pattern in the equations of the second, third and fourth terms. Also the equations can also be written as shown in the right hand sides of the equations.

$f(2) = 2(1) + 1 = 3 = 2^{2} \ – \ 1$

$f(3) = 2(3) + 1 = 7 = 2^{3} \ – \ 1$

$f(4) = 2(7) + 1 = 15 = 2^{4} \ – \ 1$

So, the closed-form of the default recursive equation is:

$f(n) = 2^{n} \ – \ 1$

The calculator uses this technique to compute the Recursive equation solution.

## Solved Examples

The following examples are solved through the Recursive Sequence Calculator.

### Example 1

The recursive relation is given as follows:

$f(n) = f(n-1) \ – \ n$

The first term for the above recursive relation is specified as follows:

$f(1) = 4$

Calculate the closed-form solution or the recurrence equation solution for the above recursive relation.

### Solution

The user must first enter the recursive relation and the first term in the input window of the calculator as given in the example.

After entering the input data, the user must press “Submit” for the calculator to process the data.

The calculator opens an output window which shows two windows.

The Input window shows the recursive relation and the first term of a particular sequence as follows:

$f(n) = f(n \ – \ 1) \ – \ n$

$f(1) = 4$

The Recurrence equation solution shows the resulting closed-form equation as follows:

$f(n) = 5 \ – \ \frac{1}{2} n(n + 1)$

### Example 2

Calculate the Recurrence equation solution for the recursive relation given as:

$f(n) = 2 f(n \ – \ 1) + n \ – \ 2$

The first term specified for the recursive equation is as follows:

$f(1) = 1$

### Solution

The user must first enter the recursive relation in the input block against the title “$f(n)$”. The recursive relation should be entered as shown in the example.

The closed-form solution requires the first term for the particular sequence. The first term is entered in the input block against the title “$f(1)$”.

The user must press “Submit” after entering the input data.

The calculator processes the input and displays the output in the following two windows.

The Input window allows the user to confirm the input data. It shows both the recursive relation and the first term as follows:

$f(n) = 2 f(n \ – \ 1) + n \ – \ 2$

$f(1) = 1$

The Recurrence equation solution window shows the closed-form solution of the recursive relation. The calculator computes the first four terms and observes a similar pattern in the four equations.

The calculator shows the result as follows:

$f(n) = 2^{n} \ – \ n$