Consecutive Integers

Consecutive integers are those integers that follow each other in a regular counting pattern. While listing consecutive integers in a sequence, no numbers are skipped in between and that is the reason why the difference between them is always fixed. The consecutive integers are integers that follow each other in increasing order. The consecutive integers formula helps in finding such integers for any given number or to check whether a set of integers are consecutive or not.

In this article, we will explore the concept of consecutive integers, consecutive even and odd integers with examples, and their formula properties. We shall also learn to determine the sum of three consecutive integers and consecutive positive integers. We will also go through some solved examples for a better understanding of the concept.

Whenever we number or count items in a sequence, we use consecutive integers. In other words, consecutive integers are integers that follow each other in a sequence with a difference that is fixed. For example, if we take the list of natural numbers, 1,2,3,4,5,6, we see that there is a difference of 1 between each integer. Similarly, we can make a list of consecutive even integers, consecutive odd integers, and many such combinations. The only point that needs to be remembered is that the difference between the integers is fixed and since they are integers, they can be positive, negative, or zero, but they do not include fractions or decimals.

We know that even numbers are multiples of 2. So, if we list the set of even integers in ascending order, they can be written as -4, -2, 0, 2, 4, 6, 8, 10, and so on. We can observe that the difference between each successive integer is 2. Thus, even consecutive integers have a difference of 2 between each predecessor and successor. For example, 6 - 4 = 2, and 4 - 2 = 2. So, if x is an even integer, then the sequence of consecutive even integers can be written as x, x + 2, x + 4, x + 6,...

We know that odd numbers are those numbers that are not divisible by 2. So, if we list the set of odd integers in ascending order, they can be written as 1, 3, 5, 7, 9, and so on. We can observe that the difference between each successive integer is 2. Thus, odd consecutive integers have a difference of 2 between each predecessor and successor. For example, 3 - 1 = 2, and 7 - 5 = 2. So, if x is an odd integer, then the sequence of consecutive odd integers can be written as x, x + 2, x + 4, x + 6,...

Using the definition of consecutive integers as discussed in the previous sections, we conclude that the consecutive integers are of form: x, x + 1, x + 2, x + 3,..., where, x is an integer, and x + 1, x + 2, .. are successive consecutive integers in sequence. In a problem involving consecutive integers, we assume the first integer to be x and the subsequent integers can be obtained by adding 1 to the previous integer.

We know that two consecutive even integers (or) two consecutive odd integers differ by 2. So any two consecutive even integers (or) consecutive odd integers are of the form: x, x + 2, x + 4, ..., where, x is an even/odd integer, and x + 2, x + 4, .. are successive even/odd consecutive integers in sequence.

Consecutive integers are those integers that follow each other in ascending order. Let us note the properties of consecutive integers.

• The difference between each consecutive integer in a sequence is the same. For example, let us observe the following list: -2, -1, 0, 1, 2. We can see that the difference between each successive number is 1. Therefore, if we take the first integer to be 'x', the series can be written as, x, x +1, x + 2, x + 3, and so on.
• The difference between each consecutive odd integer is always 2. For example, if we make a list of consecutive odd integers, -3, -1, 1, 3, 5, 7, and so on, we see that the difference between each successive integer is 2. (7 - 5 = 2)
• The difference between each consecutive even integer is always 2. For example, if we make a list of consecutive even integers, -4, -2, 0, 2, 6, 8, 10, 12, and so on, we see that the difference between each successive integer is 2. (10 - 8 = 2)
• The sum of 'n' consecutive odd integers is always divisible by 'n'. For example, if we take the sum of any 2 consecutive odd numbers, then their sum will always be divisible by 2. Similarly, the sum of any 15 consecutive odd numbers is always divisible by 15.

Consecutive positive integers are a sequence of natural numbers with a fixed difference. For example, 1, 2, 3, 4, 5,... are consecutive positive integers with a fixed difference equal to 1. We can have various sequences of consecutive positive integers such as consecutive even positive integers and consecutive odd positive integers. Let us solve an example related to the concept for a better understanding.

Example: Find two consecutive positive integers sum of whose squares is 365.

Solution: Assume one integer to be x, then the other integer is x + 1, as the difference between two consecutive positive integers is 1.

We have x² + (x + 1)² = 365

⇒ x² + x² + 1 + 2x = 365 --- [Using algebraic identity (a + b)² = a² + 2ab + b²]

⇒ 2x² + 2x + 1 = 365

⇒ 2x² + 2x + 1 - 365 = 0

⇒ 2x² + 2x - 364 = 0

⇒ x² + x - 182 = 0

⇒ x² + x - 182 = 0

⇒ x² + 14x - 13x - 182 = 0

⇒ x(x + 14) - 13 (x + 14) = 0

⇒ (x - 13) (x + 14) = 0

⇒ x = 13, or x = -14

Since, we need a positive integer, x = -14 is rejected. So, x = 13.

Then, x + 1 = 14

Answer: The required consecutive positive integers are 13 and 14.

Three consecutive integers are a sequence of three integers such that their difference is fixed. Generally, we find three consecutive integers given with a certain condition to solve problems based on consecutive integers. Let us solve an example to understand this better.

Example: Find three consecutive integers such that their sum is 51.

Solution: Assume the first integer to be x, then the other two integers are x + 1 and x + 2.

We have x + (x + 1) + (x + 2) = 51

⇒ x + x + 1 + x + 2 = 51

⇒ 3x + 3 = 51

⇒ 3(x + 1) = 3 × 17

⇒ x + 1 = 17

⇒ x = 17 - 1

⇒ x = 16

So, the other two integers are 16 + 1 = 17 and 16 + 2 = 18.

Answer: The required integers are 16, 17, and 18.

Important Notes Section on Consecutive Integers

• Consecutive integers are integers that follow each other in a sequence with a difference that is fixed.
• Consecutive integers can consist of positive, negative integers, and zero.
• Consecutive even and odd integers have a fixed difference of 2.

Related Articles

• Formula for Adding Consecutive Numbers
• Consecutive Numbers

What are Consecutive Integers?

Consecutive integers are those integers that are listed in a regular counting pattern. While listing consecutive integers in a sequence, no numbers are skipped in between and that is the reason why the difference between them is always fixed. For example, consecutive integers can be listed as -4, -3, -2, -1, 0, 1, 2, 3, and so on, where the difference between each integer is 1.

What is the Consecutive Integers Formula?

The consecutive integers formula is expressed in an easy way. If 'x' is the first consecutive integer, then the second consecutive integer will be x + 1, the third one will be x + 2, and so on. So, if we substitute an integer in this formula, we will get a series of consecutive integers. For example, if we substitute 3 as the value of 'x', the series of consecutive integers will be 3, 4, 5, and so on.

What are Odd Consecutive Integers?

Odd consecutive integers have a difference of 2 between each predecessor and successor. For example, if we write a series of odd consecutive integers as 7, 9, 11, 13, 15, and so on, we can see that there is a fixed difference of 2 between each successive number. For example, 11 - 9 = 2, and 9 - 7 = 2.

What are Three Consecutive Integers?

Three consecutive integers mean three numbers that are written in a regular counting pattern like 1, 2, 3, and so on. We can also write a list of even consecutive integers like 2, 4, 6, or odd consecutive integers like 5, 7, 9. The only thing to be kept in mind is that the difference between these consecutive integers should be fixed.

How to Find two Consecutive Integers?

We can find two consecutive integers using the consecutive integers formula. If we assume 'x' to be the first consecutive integer, then the second consecutive integer will be x + 1, the third one will be x + 2, and so on. So, if we substitute an integer in this formula, we will get a series of consecutive integers. For example, if we substitute 5 as the value of 'x', the series of consecutive integers will be 5, 6, 7 and so on.

How to Find the Sum of Consecutive Integers?

If we know the list of consecutive integers, we can easily find their sum by adding them. For example, if we need to find the sum of the first three even consecutive integers, we will list them as, 2, 4, 6. The sum of these consecutive integers will be 2 + 4 + 6 = 12.

Can Consecutive Integers be Negative?

Yes, consecutive integers can be negative because integers include positive numbers, zero, and negative numbers. For example, the following series shows negative and positive consecutive integers: -3, -2, -1, 0, 1, 2, 3 ,4. It should be noted that there is a fixed difference of 1 between each consecutive integer.

Can Consecutive Integers be Odd?

Yes, consecutive integers can be odd. For example, the odd consecutive integers can be listed as 1, 3, 5, 7, 9, and so on. Here, the difference between each consecutive integer is 2.