## Concept Understanding

*photo credit: 12:30 / Pexels*

In mathematics, every now and then we come across the word consecutive numbers. Consecutive number, actually, are the numbers that follow each other in the ascending order, i.e., smallest to largest. Consecutive numbers can mean the numbers following continuously or in an unbroken sequence. For example, if you write the numbers continuously starting from 1 and continue forward without breaking the sequence then the numbers are consecutive.

The may be a set of integers the mean and median where are equal. For example if x is an integer 10 X + 1 and X + 2 are the consecutive integers.

Examples

4, 5, 6, 7, 8 …..

-1, 0, 1, 2, 3, 4 …

2, 4, 6, 8, 10 …

Etc.

From the above examples we can see that the consecutive numbers follow a sequence. The difference between the preceding and succeeding numbers always is same. The list of ordinal number from 1 to 100 can be learned easily and is quite helpful for specifying the order of any given object

## Types of Consecutive Numbers

Consecutive odd numbers The numbers are ordered as 1, 3, 5, 7, 9, 11, and 13 ……so on when written from the smallest to the largest are called consecutive odd numbers.

Consecutive even numbers The numbers are ordered as 2, 4, 6, 8, 10, 12 ………so on when written from the smallest to the largest forms the list of consecutive even numbers.

These are called consecutive odd and even numbers because the difference between any predecessor-successor pair is 2.

## Properties of Consecutive Numbers

- The difference between any two consecutive odd or even numbers is 2.
- If m is an odd number, then the total sum of m consecutive integers will be divisible by m.
- Depending upon the set which has been started there might be too even numbers and one odd number or vice versa in a set of three consecutive integers.
- There will be accurately one number divisible by n in any set of n consecutive numbers.
- The difference between the predecessor and the successor of any number in a sequence is always the same.

## Consecutive number formula

For a number n, the *next two consecutive numbers = (n + 1) and (n + 2).* Given below are more consecutive number formulas.

The formula for *adding n consecutive numbers = [a + (a + 1) + (a + 2) + …. {a + (n-1)}].* So, the sum of n consecutive numbers or sum of n terms of AP (Arithmetic Progression) = (n/2) × (first number + last number).

Even Consecutive Numbers Formula = 2n, 2n+2, 2n+4, 2n+6 …

Odd Consecutive Numbers Formula = 2n+1, 2n+3, 2n+5, 2n+7 …

Consecutive Even and Odd Integers:

We can also have consecutive even and odd integers.

Example: Consecutive Even Integers: – 8, –6, –4, –2, 0, 2, 4, 6 …..

Example: Consecutive Odd Integers: –9, –7, –5, –3, –1, 1, 3, 5, 7 ….

## Divisibility Of Product Of Consecutive Integers

In a sequence of consecutive numbers, the product will always be divisible by the number of integers.

If there is an even number of integers in a consecutive integer sequence, the product will always be divisible by 2.

Overall, the product of n consecutive integers is divisible by n!

N! Is “n factorial.” For a positive integer n, n! Is the product of all positive integers less than or equal to n.

Example: 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

The term consecutive numbers is often used to frame word problems.

### How do you find Consecutive Numbers?

Write the given numbers in order from the smallest to the largest. Then find the difference between any predecessor-successor pair. The missing number in the consecutive number list is predecessor + difference

## Summing up

This is the perfect example of consecutive numbers. The list actually is a pattern that makes us understand the concept of consecutive numbers in an efficient manner. So the explanation and examples above help you to understand about consecutive numbers, the word you might have heard in your day to day life too.

Cuemath adds to the theory sets of integers, continuous whole numbers etc.

### Fun Facts

The sum of any two consecutive numbers is always odd. Example, 4 + 5 = 9; –8 + (–7) = –15.