Contents

- Definition
- Successive Numbers and Numerical Analysis
- Successive Numbers in Calculus
- Successive Numbers and Optimization
- Generation of Successive Numbers Using Iterative Techniques
- Generation of Successive Numbers Using Recursive Methods
- Generation of Successive Numbers Using Convergence Criteria
- Example of Successive Values

# Successive|Definition & Meaning

## Definition

Successive literally means to come after the other in some discernable order. For example, the days of a week are successive. Consecutive is not the same as successive. 1, 2, 3, and 4 are both successive and consecutive natural numbers because there are no gaps. However, 1, 3, 5, and 7 are only successive as there is a gap between them, but they still follow an order (+2).

In mathematics, successive meansÂ a **sequence** of values created in a particular **order**, typically via a mathematical **procedure** or **algorithm**. The sequence is **generated** by **continually** performing a mathematical **function** to a **beginning** value, **yielding** a new value with each **iteration**.

The mathematical approach utilized to **construct** the sequence **determines** the **ordering** of the numbers. It is common practice to utilize **successive** integers to **estimate** solutions to issues that **cannot** be addressed **exactly** or to determine the **maximum** valueÂ or **minimum** valueÂ of a function.

## Successive Numbers and Numerical Analysis

It is commonÂ practiceÂ in **numerical** analysis to employ successive numbers to **approximately** solve issues that could notÂ be solved **precisely**. Finding the **roots** of problems and **resolving** differential equations are two **frequent** uses of **successive** numbers in numerical analysis.

### Finding Roots

Finding theÂ **roots** of an integerÂ is one of the most **popular** uses of **successive** values in numerical analysis. An equation’s root is a number that, when **introduced** into the **equation**, causes it to reach **zero**. For instance, the roots for the equation **below** will be **x = 3**Â and **x = -3**.

f(x) = x^{2} – 9Â = 0

Numerical techniques like the **bisection** method, **secant** method, and **Newton-Raphson** method are frequently employed to **discover** approximations of equations’ **roots** because it is usually **impossible** to calculate the equation’s rootsÂ **analytically**. These **techniques** work by **repeatedly** performing mathematical **computations**Â on an **initial** estimate to get **repeated** estimations of the root.

### Solving the Differential Equations

The **solution** of differential equations is yet another **significant** use of successive numbers in **numerical** analysis. **Differential** equationsÂ areÂ mathematical **expressions** thatÂ explain how a **phenomenon** alters over time. They are employed in the **modeling** of a vast array of **physical** and **biological** processes. one popular approach for resolving differential equations isÂ the **Euler method**. This method is focused on **constructing** successive **estimations** of the problemsÂ by **repeatedly** performing mathematical functions on a **starting** value.

## Successive Numbers in Calculus

Numerous **calculus** operations, such as successive **derivatives** andÂ **integrals**, make use of successive numbers.

### SuccessiveÂ Derivatives

A function’s **derivative** in calculus is a **measurement** of how the function **alters** as its input alters.Â The **second** derivatives, also referred to as the **second-order** derivatives, areÂ derivativesÂ of the **first** derivatives. In the same manner, the derivative ofÂ **(n-1) ^{th}** orderÂ derivative is the

**n**orderÂ derivative. They are often referred to as

^{th}**successive**derivatives and are employed to

**examine**the

**local**behavior of a function

**close**to a point.

### SuccessiveÂ Integrals

The **integral** of functions in calculus represents the **region** beneath the **curve** of the functions. The function whose **derivative** is the **original** function is known asÂ the **definite** integral and isÂ commonly referred to as the **antiderivative**. in the same manner, the **n ^{th}** integral is the integral whose

**derivative**isÂ the

**(n-1)**integral. These can be used to determine the

^{th}**overall**variation in a function during a specific

**period of time**and are often referred to as

**successive**integrals.

## Successive Numbers and Optimization

In the process of optimizing, we determine the **maximum or minimum** value of a function by using **successive** numbers. The process of **determining** which of a number of **potential** solutions is the most **beneficial** is known as **optimization**. In a variety of optimization methods, such as gradient** descent**, **conjugate gradient**, and **Newton’s method**, successive integers play an important role.

### Minimizing and Maximizing Functions

When determining the minimum or maximum value of a function, successive numbers are considered. It is possible for the function to be quite straightforward, such as a quadratic function, or rather complicated, such as a neural network. Various optimization strategies, such as gradient descent, conjugate gradient, and Newton’s method, make use of subsequent integers to continuously improve an initial approximation of the least or maximum value.

### Linear and Non-linear Optimisation

In both **linear** and non-linear optimization, **successive** numbers are **essential**. When it comes to optimization, **linear** optimization **focuses** on linear functions and **limitations**, while **non-linear** optimization **focuses** on non-linear **functions** and limitations. The non-linear optimization process is utilized in **various** domains, including **control systems**, **image analysis**, and **computer vision**.

## Generation of Successive Numbers Using Iterative Techniques

In mathematics, the **generation** of **successive** numbers frequently makes use of a method called the **iterative** approach. Iterative approaches entail **continually** performing a mathematical **operation** to theÂ **original** value, which ultimately results in a **different** value for each **iteration** of the process. The mathematical **algorithm** that was used to **construct** the series of numbers is what decides the **order** of the **numbers** within it.

There are a **variety** of iterative approaches that can be **utilized**, such as the following, to produce successive numbers:

- Fixed-Point Iteration
- Newton-Raphson Method
- Successive Over-Relaxation
- Jacobi and Gauss-Seidel method

## Generation of Successive Numbers Using Recursive Methods

In mathematics, generating **successive** numbers can be **accomplished** through the application of **recursive** techniques. In recursive approaches, a **sequence** of numbers is **defined** in terms of the numbers that came **before** them in the sequence. This is often accomplished through the utilization of a recursive **equation** or **algorithm**.

There are a variety of recursive **approaches** that you can employÂ to produce successive numbers, such as the following:

### Fibonacci Numbers

The famous **Fibonacci** numbers are an iconic illustration of a **recursive** sequence. Each successive number in the series **represents** the **addition** of the two numbers that came **before** it. The digits 0 and 1 mark the **beginning** of the sequence, and from that **point** on, each **succeeding** number in the series is **calculated** by adding the **two** numbers that came **before** it.

### Recursive Algorithm

An algorithm is said to be recursive if it can **invoke** itself with data that is **less** complex. Computing the **factorial** of an integer, for instance, can be performed **recursively**.

### Recursive Formulas

A **formula** is said to be recursive if it **describes** a sequence of numbers by making **reference** to the numbers that came **before** them in the **sequence**.

## Generation of Successive Numbers Using Convergence Criteria

The **application** of convergence criteria allows one to **ascertain** whether or not a succession of successive numbers has, in fact, **converged** to a certain number or answer.

The **criteria** are employed to **assess** whether the sequence has **acquired** a level of **stability**, which is a state in which the values **contained** inside the sequence do notÂ varyÂ **substantially** from one **iteration** to the next.

For generating successive numbers, one can utilize any one of a number of distinct convergence criteria, such as the following:

### Absolute Error Criteria

This criterion is determined by the **absolute** deviation that exists between the **sequence’s** twoÂ most **recent** values.Â The sequence is deemed to have **converged** when the **variation** is smaller than a **predetermined** criterion for success.

### Relative Error Criteria

This criterion is **determined** by examining the **degree** of **dissimilarity** that exists between the sequence’s **two**Â most **recent** values. It is determined that the sequence convergesÂ when the **disparity** is smaller than a **particular** threshold.

### Ratio Test

This criterion **evaluates** how **subsequent** valuesÂ in the sequence relate to one another in terms of their **ratio**. When the ratio gets close to a **particular** value, we say that the sequence has **converged** because it has **reached** that point.

## Example of Successive Values

Complete the **series** of following **successive** values

- 1, 3, 5, ?, 9, 11
- Monday, Tuesday, Wednesday, ?, Friday, Saturday, and Sunday
- 2, 4, 6, ?, 10, 12

### Solution

In **(a),** we can clearly see that the series is of **odd **numbers. Thus, the completed series is as follows:

**(a)** 1, 3, 5, 7, 9, 11

In **(b), **we can clearly see that the series is on **weekdays,** so the completed series is as follows:

**(b)Â **Monday, Tuesday, Wednesday, **Thursday**, Friday, Saturday, Sunday

In **(c), **we can clearly see that the series is of **even **numbers, so the completed series is as follows:

**(c)** 2, 4, 6, 8, 10, 12

*All images/mathematical drawings were created with GeoGebra.*