Consecutive Numbers in Maths

Consecutive Numbers are the numbers that are next to each other in a queue. These numbers always consist of a difference of 1. The first step to understanding consecutive numbers is to be familiar with the number system.

In mathematics, numbers that sequentially follow one another from smallest to largest are known as consecutive numbers. This means that each number in the sequence is followed immediately by the next number without any gaps. For example, the consecutive numbers from 1 to 8 are 1, 2, 3, 4, 5, 6, 7, and 8.

In this article, we will understand the definition of consecutive numbers in maths followed by some basic examples, consecutive numbers from 1 to 100, their types, properties, formulas, how to find consecutive numbers, etc.

What are Consecutive Numbers?

Consecutive Numbers in maths are those numbers that are present in a sequential manner in increasing order. We can also understand consecutive numbers through the concept of predecessor and successor.

The number preceding another number is called its predecessor, while the number following it is called its successor.

Consecutive Numbers Meaning

The numbers that follow one another in ascending order are called consecutive numbers. In other words, we can say that the numbers which have the same difference between them are known as consecutive numbers.

Consecutive Numbers Examples

Suppose there is a series of numbers i.e. 1, 2, 3, 4, 5, 6 . . . These numbers have the same difference between them. Each number has a difference of 1. So these all are consecutive numbers.

According to another concept, if we took the number "3" from this series. The predecessor of 3 is 2 while its successor is 4. Here the difference between 3-2 = 1 and 4-3 = 1 is also the same. In this way, it is also proved that these are consecutive numbers.

The idea of successor and predecessor is very important in finding Consecutive Numbers. So now let's begin with Successor and Predecessor.

Successor and Predecessor

Idea of successors and predecessors, helps us understand numbers that come one after the other. Let's take a look at natural numbers as an example.

Natural numbers are 1, 2, 3, 4, 5, 6,...

• The successor of a number is the number that is one greater than the previous number, i.e. successor of n is n+1
• The predecessor of a number is the number that is one lesser than the next number, i.e. predecessor of n is n-1

Thus, consecutive numbers are found using the concept of successor.

Consecutive Numbers 1 to 100

Consecutive Numbers refers to the numbers that are written from 1 to 100 continuously. The chart below shows consecutive numbers from 1 to 100:

Types of Consecutive Numbers

In this section, we will learn about different kind of consecutive numbers. Consecutive numbers consists some types which helps us to differentiate. These are classified mainly into 3 categories:

• Natural Consecutive Numbers
• Even Consecutive Numbers
• Odd Consecutive Numbers
• Consecutive Integers

Consecutive Natural Numbers

To understand consecutive natural numbers, we have to understand natural numbers first. We use natural numbers for normal counting.

In simple words, all whole numbers are natural numbers except zero (0). Natural consecutive numbers are those arranged in ascending order from small to large numbers.

For example: The natural consecutive numbers from 1 to 10 are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

Consecutive Even Numbers

Even numbers are the numbers which can be divided by 2. So the even numbers in which there is a difference of 2, are considered as Even Consecutive Numbers.

For example: The even consecutive numbers from 1 to 20 are: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20. In this sequence the difference between each number is 2 and they all can be divided by 2, so these are called Consecutive Even Numbers.

Consecutive Odd Numbers

Odd numbers are those numbers which can't be divide by 2. The odd consecutive numbers also have a difference of 2 between them but they can't be divisible by 2. This is the main difference between even and Odd Consecutive Numbers.

Consecutive Integers

In consecutive numbers, we can only take numbers starting from 1. Consecutive integers are those integers which have both negative and positive set of numbers of numbers scale. These are of two types:

• Positive consecutive integers: 0, 1, 2, 3, 4, 5, 6.....
• Negative consecutive integers: -1, -2, -3, -4, -5, -6....

Properties of Consecutive Numbers

In this section, we will learn some basic properties of consecutive numbers which will help us to deeply understand the topic and we can easily solve the problems related to consecutive numbers.

• There should always be a fixed difference between any two consecutive numbers. Any two odd consecutive numbers will always have a difference of 2. Similarly, any two even consecutive numbers will also have a difference of 2.
• If two odd consecutive numbers are added, their addition will always be an even number. Similarly when any two even consecutive numbers are summed up, they will always give an even number. While adding two consecutive numbers, it will always give an odd number.
• If a sequence is made up of odd numbers, their sum will always be divisible by the total number 'n'
• HCF of Two Consecutive Numbers is always 1.

HCF of Two Consecutive Numbers

HCF means "Highest Common Factor". The greatest factor which is common between two or more than two numbers is called HCF. HCF is the common factor that can divide those numbers.

HCF of two consecutive numbers is always 1 as one(1) is the only common factor between any two consecutive numbers.

For example: A series of two consecutive numbers is given as 56, 57.. In this series the common factor between 56 and 57 is only 1. Therefore the HCF of two consecutive numbers 56 and 57 is 1.

Consecutive Number Formulas

To find the consecutive numbers, some formulas are used. These formulas help us to find the series of consecutive numbers.

Consecutive Integers Formula

As we know that, integers always follow a sequence and the difference between two consecutive integers is always equal to 1.

• For any integer n, the formula to get a consecutive integer is (n + 1)

Natural Consecutive Number Formula

For natural consecutive numbers, suppose 'a' is an integer. Now, to find the next numbers, simply use the formula a+1, a+2, a+3....In this way, we can easily find an order of numbers. Suppose 'a' is given as 6. Now add 6+1= 7, 6+2= 8, 6+3= 9...So, the consecutive number sequence is 6, 7, 8, 9.....and so on. Just keep adding consecutive numbers in the original number.

Natural Consecutive Numbers = a, a+1, a+2 . . .

Consecutive Even Number Formula

Similarly, we represent consecutive even integers as '2a'. To find the next even consecutive numbers, simply add even consecutive numbers to 2a as 2a+2, 2a+4, 2a+6 and so on because even consecutive numbers have a difference of 2. Now, if the value of 'a' is 7 then 2a= 2×7= 14. Now add 14+2= 16, 14+4= 18, 14+6= 20 etc. The even consecutive series is 14, 16, 18, 20....and so on.

Even Consecutive Numbers: 2a, 2a+2, 2a+4 . . .

Consecutive Odd Number Formula

While in the case of consecutive odd integers, we represent it as '2a+1'. Now, the next odd consecutive integers will be 2a+3, 2a+5, 2a+7 etc. due to the difference of 2. Here, we take the value of 'a' is 3 which will give 2×3+1 = 7. Now kept adding to find next odd numbers as 2×3+3 = 9, 2×3+5 = 11, 2×3+7 = 13.... and so on.

Odd Consecutive Numbers: 2a+1, 2a+3, 2a+5 . . .

The formulas for consecutive numbers are summarized below:

Sum of Consecutive Numbers

We have studied earlier that a consecutive number always differs by the same value i.e. 1. Now, in this section, we will learn about performing the addition of two or more than two consecutive numbers.

To find the sum of consecutive numbers, a simple formula is used. The formula to add 'n' number of consecutive integers is:

Sum of 'n' consecutive numbers= n/2 ( first term + last term)

where,

• n is count of number of consecutive terms under consideration
• n = last term - first term + 1

Example: Find the sum of consecutive integers from 20 to 30.

Solution:

Sum of n consecutive numbers= n/2 ( 1st term + last term)

here, first term is 20 and last term is 30. First find n= 30-20+1= 11. Now, use the formula:

Sum of Consecutive Integers= 11/2 (20+30) = 11/2(50)= 11×25= 275.

Let's learn Sum of Consecutive Numbers Formula for different numbers.

Sum of Consecutive Natural Numbers

The formula for calculating the addition of consecutive natural numbers is:

Sum of Consecutive Natural Numbers= n(n+1)/2

Here,

• n = last natural number
• n+1 = next natural number

Example: Determine the sum of first 10 natural numbers. The first 10 natural numbers are (1, 2, 3, 4, 5, 6 ,7 ,8 9, 10).

Solution:

The last natural number is '10'.

sum of consecutive natural numbers= (10(10+1))/2= (10×11)/2= 110/2= 55.

Sum of Consecutive Even Numbers

To evaluate the sum of consecutive even numbers, the formula is:

Sum of Even Consecutive Numbers = n(n+1)

Example: Calculate the sum of first 3 even consecutive numbers.

Solution:

Given even term is '3'. The first 3 even consecutive numbers are 2,4 and 6.

sum of even consecutive numbers= 3(3+1)= 3×4= 12.

Sum of Odd Consecutive Numbers

The formula for finding the addition of consecutive odd numbers is:

Sum of Odd Consecutive Numbers = n²

Example: Find out the sum of first 6 odd consecutive numbers.

Solution:

The value of 'n' is '6'. The first 6 odd consecutive numbers will be (1, 3, 5, 7, 9, 11).

sum of consecutive odd numbers = (6)² = 36.

How to Find Consecutive Numbers when Sum is Given?

To find consecutive numbers when sum is given is explained in the example added below,

Example: Find three consecutive numbers when the sum of these three numbers is 24.

Let, the three consecutive numbers are, (n -1), n, (n + 1)

Given, sum of number = 24

(n -1) + (n) + (n + 1) = 24

n + n + n - 1 + 1 = 24

3n = 24

n = 24/3 = 8

Thus, the three consecutive number are,

• n - 1 = 8 - 1 = 7
• n = 8
• n + 1 = 8 + 1 = 9

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Sample Problems on Consecutive Numbers

Here are some sample examples related to consecutive numbers:

Example 1: Find the value of integers, if the sum of two consecutive integers is 41?

Solution:

Let two consecutive numbers be z and z+1

Given that:

z+ (z+1)= 41

2z + 1 = 41

2z = 41-1

z = 40/2

z = 20 and z+1= 20+1= 21

Hence, the consecutive integers whose sum is 41 are 20 and 21.

Example 2: If two consecutive odd numbers have a sum of 92. Find the numbers?

Solution:

Suppose two consecutive odd numbers are n and n+2 since the odd consecutive numbers differ by 2.

Given that:

n+ (n+2)= 92

⇒ 2n+2= 92

⇒ 2n= 92-2

⇒ n= 90/2

⇒ n= 45 and n+2= 45+2= 47

Therefore, the two consecutive odd numbers whose sum is 92 are 45 and 47.

Example 3: Find the Sum of First 50 Natural Numbers.

Solution:

Since, n = 50

We know that sum of first 'n' natural numbers = n(n + 1)/2

Hence, Sum of First 50 natural numbers (50 × 51)/2 = 25 × 51 = 1275

Example 4: Find three consecutive integers such that the sum of the second and third integers is 31.

Solution:

Let the three consecutive integers be x, x+1, and x+2.

Given that the sum of the second and third integers is 31: (x+1)+(x+2)=31

Simplifying: 2x+3=31

Subtract 3 from both sides: 2x=28

Divide by 2: x=14

So the three consecutive integers are x=14, x+1 = 15, and x+2=16.

Example 5: The sum of four consecutive even numbers is 100. Find the numbers.

Solution:

Let the four consecutive even numbers be y, y+2, y+4, and y+6.

Given that the sum of these numbers is 100: y+(y+2)+(y+4)+(y+6)=100

Simplifying: 4y+12=100

Subtract 12 from both sides: 4y=88

Divide by 4: y=22

So the four consecutive even numbers are y=22, y+2=24, y+4=26, and y+6=28.

Practice Problems on Consecutive Numbers

After understanding consecutive numbers and learning its important properties and formulas, now it's time to practice some questions of consecutive numbers to strengthen the topic more. These are few practice questions which will help you to revise your concept:

Problem 1: If the product of any four consecutive numbers is 32,760. Find those numbers?

Problem 2: Jenny writes 3 consecutive numbers on a paper. The sum of these is 150. What are the numbers that Jenny wrote on the paper?

Problem 3: Find the even consecutive numbers starting from 50 to 70?

Problem 4: The HCF of two consecutive odd numbers is 45. Solve the question and find the numbers.

Problem 5: Three consecutive even numbers are given in which the sum of the first two is 26. Find those numbers?

Problem 6: The sum of four consecutive odd numbers is 80. What are those numbers?

Problem 7: Find three consecutive multiples of 7 whose sum is 168.

Problem 8: The difference between the square of two consecutive even numbers is 36. What are those numbers?

Problem 9: The sum of the first and last numbers of five consecutive numbers is 62. Find the numbers.

Problem 10: If the sum of three consecutive odd numbers is 75, what are the numbers?

Conclusion of Consecutive Numbers in Maths

Consecutive numbers play a fundamental role in mathematics, providing a simple yet powerful way to understand sequences and numerical patterns. These numbers, which follow each other in order without gaps, are used in various mathematical concepts, including arithmetic operations, algebra, and number theory. Recognizing and working with consecutive numbers help in solving problems related to series, progressions, and even real-life situations.