# What is the difference between recursive and explicit?

The main difference between recursive and explicit is that a recursive formula gives the value of a particular term based on the previous term, while an explicit formula gives the value of a particular term based on position.

A sequence is an important concept in mathematics. It refers to a series of numbers arranged in an order. We can represent an arithmetic sequence with a formula. In other words, we can calculate each term in the sequence directly with a formula. There are two types of formulas as recursive and explicit formulas. A formula describes a way to find any term in the sequence.

### Key areas covered

1. What is recursive - definition, functionality 2. What is explicit - definition, functionality 3. Difference between recursive and explicit - comparison of the main differences

### key terms

Explicit formula, recursive formula

## What is recursive?

In a recursive formula, we can find the value of a particular term based on the previous term.

For example, assume a formula like this.

a (n) = a (n-1) +5

The first term of the sequence is a (1) = 3

The second term is as follows.

a (2) = a (2-1) + 5

a (2) = a (1) + 5

We can replace the above formula with the value. Then it gives the result for a (2).

a (2) = 3 + 5

a (2) = 8

Similarly, we can find the third term as follows.

a (3) = a (2) + 5

a (3) = 8 + 5 = 13

The calculation of the fourth term is as follows.

a (4) = a (3) + 5

a (4) = 13 + 5 = 18

We can also calculate the values ​​of the terms in the sequence. To find a (4) we need the value of a (3). To find a (3) we need the value of a (2) and to find the value a (2) we need the value of a (1). Therefore, the previous term or terms is required to determine the value of a particular term. That is the functionality of recursive formulas.

## What is explicit?

In explicit formulas, we can determine the value of a particular term based on its position.

Assume a formula like this.

a (n) = 2 (n-1) + 4

The first term is as follows.

a (1) = 2 (1-1) + 4 = 0 + 4 = 4

The second term is as follows.

a (2) = 2 (2-1) + 4 = 2 + 4 = 6

The third term is as follows.

a (3) = 2 (3-1) + 4 = 4 +4 = 8

The fourth term is as follows.

a (4) = 2 (4-1) + 4 = 8 + 4 = 12

We can also find the values ​​of each term in the sequence.

When looking at the sequence it can be seen that it is possible to calculate the value of a certain term based on the position. This is how an explicit formula works.

## Difference between recursive and explicit

### definition

For a sequence a 1 , a 2 , a 3 ... a n , a recursive formula is a formula that requires computing all previous terms to find the value of a n . For a sequence a1, a2, a3… a n , the explicit formula is a formula that can calculate the value of a n based on its position. So this is the main difference between recursive and explicit.

### Functionality

In a recursive formula, we can find the value of a term in the sequence based on the value of the previous term. However, in an explicit formula, we can find the value of a term in the sequence based on its position. Hence, this is another difference between recursive and explicit.

### diploma

We can represent a sequence with a formula. A formula can be either recursive or explicit. The main difference between recursive and explicit is that the recursive formula gives the value of a particular term based on the previous term, while the explicit formula gives the value of a particular term based on position.

##### Reference:

1. “Recursive Formulas for Arithmetic Sequences.” Khan Academy, Khan Academy, Available here . 2.Mathwords: Removable Discontinuity, available here . 3. “Explicit Formulas for Arithmetic Sequences.” Khan Academy, Khan Academy, Available here .

##### Image courtesy:

1. “Random Mathematical Formulas Illustrating the Field of Pure Mathematics” By Wallpoper (Public Domain) via Commons Wikimedia

### About the author: Lithmee

Lithmee has a Bachelor of Science in Computer Systems Engineering and is currently studying for her Masters in Computer Science. She is passionate about sharing her knowledge of programming, data science and computer systems.