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