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All you do when you begin counting natural numbers is count numbers in a linear form. Numbers that are in an ordered sequence after one another are called consecutive integers. Essentially, consecutive integers are what you get when you count any number in a set by one. These are represented by the numbers n, n +1, n + 2, n + 3,…, where n is an integer. Consecutive integers are the numbers that follow in a fixed sequence, with each number being one more than the number before it.In this article, let’s learn more about consecutive integers, about it’s types, formulas and more along with some examples.
Table of Contents
What are Consecutive Integers?
Consecutive integers are used whenever we number or count things in a sequence. Stated differently, consecutive integers are sets of numbers that are arranged in a definite order after one another. Take the list of natural numbers 1, 2, 3, 4, 5, and 6, for instance. We can see that each integer differs by 1. In the same manner, a list of consecutive odd numbers, consecutive even numbers, and numerous other combinations can be created. All that has to be kept in mind is that the integers are different from each other in a definite way and might be positive, negative, or zero; decimals and fractions are not included.
Examples of Consecutive Integers
A few examples of consecutive integers are as follows for better understanding:
- 0, 1, 2, 3, 4, 5, 6, 7,…..
- -20, -19, -18, -17, -16,…..
- 101, 102, 103, 104, 105,….
- -4, -3, -2, -1, 0, 1, 2,….
Consecutive Even Integers
We know that all even numbers are multiples of 2. Thus, the set of even numbers can be expressed as -4, -2, 0, 2, 4, 6, 8, 10, and so on if we list them in ascending order. It is clear that there is a difference between each consecutive integer. As a result, there is a difference of 2 between every predecessor and successor in even consecutive integers. For instance, 4 – 2 = 2 and 6 – 4 = 2 are examples. Thus, if x is an even number, the series of even numbers that occur can be expressed as follows: x, x + 2, x + 4, x + 6,…
Also Read: Rational And Irrational Numbers: Differences, Examples
Consecutive Odd Integers
Those numbers that are not divisible by two are known as odd numbers. Thus, the set of odd numbers can be expressed as 1, 3, 5, 7, 9, and so on if we list them in ascending order. It is clear that there is a difference between each consecutive integer. As a result, there is a difference of 2 between each predecessor and successor for odd consecutive integers. Take 3 – 1 = 2, and 7 – 5 = 2, as examples. Thus, if x is an odd number, the series of odd numbers that follow can be expressed as follows: x, x + 2, x + 4, x + 6,…
Consecutive Integers Formula
Utilizing the idea of fundamental algebraic expressions, the formula for finding the consecutive integers is extremely simple.
The formula for the consecutive integer for each given integer x is x + 1.
For Even Consecutive Integers: The formula to find the even consecutive integer is 2x.
For Odd Consecutive Integers: The odd consecutive integer can be found using the formula 2x +1.
Three Consecutive Integers
A set of three integers with a fixed difference is referred to as three consecutive integers. Typically, we find three consecutive integers given specific criteria to solve problems. In order to better understand this, let’s solve an example.
As an example: find three consecutive integers with a sum of 51.
Solution,
Let the first integer 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.
The required integers are 16, 17, and 18.
Also Read: What are Co Prime Numbers?
Properties of Consecutive Integers
Consecutive integers have some special properties which need to be kept in mind when talking about it. This will help us understand consecutive integers in a better way. Let’s check them below:
- The difference between any two consecutive numbers is always one (1).
- The difference of two consecutive even numbers is always equal to 2.
- The difference between any two consecutive odd numbers is always 2.
- A three-consecutive integer product, excluding 0, is always divisible by 6.
Solved Examples of Consecutive Integers
To understand the consecutive integers and to know how to find them, let us see some of the examples mentioned below:
- Find three consecutive integers after 77.
Solution,
Three consecutive integers sequences after x will be x + 1, x + 2, and x + 3
So, three consecutive integers after 77 are,
= 77 + 1, 77 + 2, and 77 + 3
= 78, 79, and 80
Hence, the three consecutive integers after 77 will be 78, 79 and 80.
- Find the set of three consecutive integers whose sum is 78.
Solution,
Using the formula of consecutive integers: x, x + 1, and x + 2.
Their sum is given to be 78. So we get the equation:
x + (x + 1) + (x + 2) = 78
3x + 3 = 78
3x = 78 – 3
3x = 75
X = 75/3
x = 25
So, the three consecutive integers are:
x = 25
x + 1 = 25 + 1 = 26
x + 2 = 25 + 2 = 27
The required integers are 25, 26, and 27.
- Suppose the sum of four consecutive odd integers is 184. Find the smallest number.
Solution,
Let x, x+2, x+4 and x+6 be the four consecutive odd integers.
Now,
x + x + 2 + x + 4 + x + 6 = 184
4x + 12 = 184
4x = 184 – 12
4x = 172
x = 172/4
x = 43
Hence, the smallest number is 43.
- If the sum of four consecutive integers is 302, then what is the product of the first and third integers?
Solution,
Lets assume four consecutive integers are x, x + 1, x + 2, x + 3
Now, the sum of four consecutive integers is 302.
Therefore, the equation becomes:
x + x + 1 + x + 2 + x + 3 = 302
4x + 6 = 302
4x = 302 – 6
4x = 296
x = 296/4
x = 74
So,
First integer is, x = 74
Third integer is, x + 2 = 76
And so, the product of the first and third integers is 74 × 76 = 5624.
FAQs
The consecutive numbers are arranged in a sequence that is continuous. The gap between the integers that come before and after it is always equal to 1.
The formula for determining consecutive integers is known as the consecutive integer formula. Assuming we start with the number n, the next integer is n + 1.
We can quickly find the sum of a list of consecutive numbers by adding them if we know the list. As an example, we would label the first three even consecutive integers as 2, 4, and 6 if we needed to figure out their sum. These numbers that follow add up to 2 + 4 + 6 = 12
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