Consecutive Numbers

Dhristi JEE 2022-24

What is the Meaning of Consecutive Numbers?

Before understanding consecutive numbers, let us look into consecutive meanings in maths. Consecutive means following each other continuously in a series or a sequence. In Mathematics, numbers that follow each other in a series are termed consecutive numbers. Consecutive numbers meaning is “The numbers which continuously follow each other in the order from smallest to largest.” 


Consecutive Numbers Examples:

Consecutive numbers from 1 to 8 are 1, 2, 3, 4, 5, 6, 7, and 8. Here, the difference between each number is 1.


Consecutive numbers from 80 to 90 with differences as 2 are 80, 82, 84, 86, 88, and 90. Here, the difference between each number is 2. This type of consecutive number system where each number is an even number and the difference between each number is 2 is called consecutive even integers which we will discuss in detail in the latter part of this chapter.


Consecutive numbers from -1 to 7 are -1, 0, 1, 2, 3, 4, 5, 6, and 7. Here, the difference between each number is 1.


What does Consecutive Natural Numbers Mean?

Natural numbers are the numbers that are used for counting and ordering. Natural numbers include all whole numbers except 0.


In the mathematical form, the natural numbers are written as follows:


N = {1, 2, 3, 4, 5, 6, 7, 8, 9, ……}


So consecutive natural numbers are the numbers that continuously follow each other in order from the smallest number to the largest number.


For example, the consecutive natural numbers from 10 to 15 are 10, 11, 12, 13, 14, and 15.


The consecutive numbers from -5 to -1 are not consecutive natural numbers because these numbers don't fall under natural numbers. These numbers are integers.


What is Meant by Consecutive Numbers with Even Integers?

An even number is one that is divisible by two and leaves a 0 as a remainder.


So Consecutive Numbers with Even integers or simply called consecutive even integers is a sequence where the numbers continuously follow each other in the order from the smallest number to the largest number with the difference between each number is 2 and each number is divisible by 2. 


Consecutive integers examples:

2, 4, 6, 8, 10, and 12 are consecutive even integers between the numbers 2 and 12.

-8, -6, -4, and -2 are consecutive even integers between the numbers -8 and -2.


What is Meant by Consecutive Numbers with Odd Integers?

A number that is not divisible by 2 is called an odd number. In the case of an odd number, the remainder is always 1. So consecutive odd integers is a sequence where the numbers continuously follow each other in the order from the smallest number to the largest number with the difference between each number is 2 and each number not divisible by 2.


Consecutive integers examples:

The consecutive odd integers between 1 and 15 are 1, 3, 5, 7, 9, 11, 13, and 15.

The consecutive odd integers between -11 and -1 are -11, -9, -7, -5, -3, and -1.


Properties of Consecutive Numbers

  • Any pair of predecessors and successors have a fixed difference. If we denote the first number as n, then n, n+1, n+2, n+3, n+4, and so on will be the consecutive numbers in the sequence. The difference between any two consecutive odd numbers is 2. For example, in the sequence of consecutive odd integers 3, 5, 7, 9, and 11, the difference between any two consecutive odd numbers is 2 i.e. 5 - 3 = 2, 7 - 5 = 2, and so on.

  • The difference between any two consecutive even numbers is 2. For example, in the sequence of consecutive even integers 12, 14, 16, 18, and 20, the difference between any two consecutive odd numbers is 2 i.e. 14 - 12 = 2, 16 - 14, = 2 and so on.

  • The sum of n consecutive numbers is divisible by n if n is an odd number. For example, let us consider a consecutive odd number sequence 5, 7, 9, 11, 13, 15, 17 which has 7 numbers in the sequence. So according to the property, the sum of this consecutive odd number sequence should be divisible by 7. 

5+7+9+11+13+15+17 = 77. 

77/7 = 11 which satisfies this property. 

  • Here, we have to note that this property applies only when n is an odd number i.e. only if the sequence is made of odd numbers.

  • The sum of two consecutive numbers is always an odd number. For example, consider a consecutive number sequence 1, 2, 3, 4, 5, 6, 7, 8, 9. Here, the sum of the two consecutive numbers will always be an odd number. 1+2 = 3, 2+3 = 5, 3+4 = 7, and so on.

  • The sum of two consecutive odd numbers is always an even number. For example, consider a consecutive odd number sequence 5, 7, 9, 11, 13, 15, 17. Here, the sum of two consecutive odd numbers will always be an even number. 5+7 = 12, 7+9 = 16, 9+11 = 20, and so on.

  • The sum of two consecutive even numbers is always an even number. For example, consider a consecutive even number sequence 12, 14, 16, 18, 20, 22, 24. Here, the sum of two consecutive even numbers will always be an even number. 12+14 = 26, 14+16 = 30, 16+18 = 34, and so on.


Problems with Consecutive Numbers

1) What is a consecutive number in maths for natural numbers between 1 and 25?

Ans: The consecutive numbers for the natural numbers between 1 and 25 are as follows:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25.


2) Write the consecutive even integers between 20 and 40. 

Ans: The consecutive even integers between 20 and 40 are as follows:

20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40.


3) Find the sum of two consecutive numbers 8 and 9.

Ans: The sum of two consecutive numbers is 8+9 = 17.


Important Concepts of Numbers and Consecutive Numbers

  • Consecutive numbers are the numbers that differ from each other by 1. Adding 1 to a number will give you the consecutive number after the original number. Similarly, subtracting 1 to a number will give you the consecutive number after the original number. 

  • The concept of consecutive numbers is useful in the chapter of arithmetic progression where each number differs its successive number by a certain difference called a common difference. 

  • For three numbers in Arithmetic progression with common difference 1, we can define these numbers as {a, a + 1, a+2}, where ‘a’ is a number.

  • The above example can also be written as {a -1, a, a+1}, where ‘a’ is a number.

  • There can be consecutive even numbers. Consecutive even numbers are the numbers that differ from their predecessor number by the ‘2’. All these numbers are also even numbers that mean they are divisible by 2 as well or they leave no remainder (or 0 remainders) when divided by 2. For example, 0, 2, 4, 6, 8, 10, 12, 14, etc. 

  • Similar to consecutive even numbers, we have consecutive odd numbers. Here, the difference between the two numbers is 2 as well but these numbers are not divisible by 2. That means these numbers will leave a remainder (remainder = 1) when divided by two. That means the numbers in a series of consecutive odd numbers will be odd in nature. The examples of odd consecutive numbers are 1, 3, 5, 7, 9, 11, 13, 15, 17, etc.


Conclusion 

  • Just remember that the sequence of a consecutive number series depends on the first number which we are representing. Say if we are representing the first number as n, then the consecutive numbers will be n+1, n+2, n+3, n+4, and so on.

  • The general form to represent consecutive even numbers is 2n where n is an integer. 

  • The general form to represent consecutive odd numbers is 2n+1 where n is an integer.

FAQs on Consecutive Numbers

1. What is a consecutive number in maths?

Numbers that continuously follow each other in the order from smallest to largest are called consecutive numbers. The difference between two consecutive numbers will always be equal to absolute ‘1’. That means the two numbers are said to be consecutive if and only if they have a unit difference. The examples of consecutive numbers are 4, 5, and 6. Here, the middle number 5 when subtracted by 1 will get us 4 (5 - 1 = 4). Similarly, when 5 is added with 1, it will yield us 6 (5 + 1 = 6). Hence, these numbers are consecutive in nature. 

2. What are consecutive even integers? 

Consecutive even integers are a sequence where the numbers continuously follow each other in the order from the smallest number to the largest number with the difference between each number as 2 and each number is divisible by 2. 

3. What is the sum of two consecutive numbers?

The sum of two consecutive numbers will always be an odd number. As these numbers will always differ by 1, they will always have one of the numbers as odd and the other one will be even. We know that the addition of one even number and one odd will always result in an odd number. For example, 10 and 11 are consecutive numbers (the difference between them is equal to 1). The sum of these two will be 21 (10+11=21). Hence, the sum of two consecutive numbers is always odd.

4. If ‘x’ is a number, what are its consecutive numbers and will they be even or odd?

If ‘x’ is a number, then its consecutive numbers will be defined as ‘x+1’ and ‘x-1’ as consecutive numbers have an absolute difference of 1. If x is an odd number, then both its consecutive numbers will be even in nature. On similar lines, if x is an even number, then both of its consecutive numbers will be odd numbers. For example, say x = 17 which is even, then 16 and 18 will be its consecutive number, both of which are even.

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