What are Consecutive Integers?
When you start counting natural numbers you are just counting the consecutive numbers or consecutive integers. Consecutive integers are integers that follow each other in a fixed sequence. Did you know that whenever you number items you are using Consecutive Integers? In fact, whenever you count by ones from any number in a set you obtain Consecutive Integers. Consecutive integers are integers that follow in a fixed sequence, each number being 1 more than the previous number, Consecutive integers are represented by n, n +1, n + 2, n + 3, ..., where n is an integer.
Examples of Consecutive Integer
For example: 23, 24, 25
Look at the following. The first set is called consecutive positive integers and the second set is called consecutive negative integers.
Example 1: 1, 2, 3, 4, 5.....
Example 2: -1, -2, -3, -4, -5, -6,.....
In the first example a set of consecutive integers is found by adding 1 to 0. You can represent the first set with this expression: n + 1, with n = 0, 1, 2, .....
The second set of consecutive integers is found by subtracting 1 from 0. You can represent the second set with this expression: 1 − n, with n = 2, 3, 4, 5,.....
Type of Consecutive Integers
There are mainly three types of consecutive integers:
Normal consecutive integers (2, 3, 4, 5, ……)
Even consecutive integers (2, 4, 6, 8, ……..)
Odd consecutive integers (3, 5, 7, 9, ………)
Even Consecutive Integers
Consecutive even integers are the set of integers such that each integer in the set differs from the previous integer by a difference of 2 and each integer is divisible by 2.
Consecutive even integers are even integers that follow each other by a difference of 2. If x is an even integer, then x + 2, x + 4, x + 6 and x + 8 are consecutive even integers.
Examples:
4, 6, 8, 10, …
-6, -4, -2, 0, …
124, 126, 128, 130, ..
You can represent consecutive even integers with the following expression: 2n + 2 with n = 0, 1, 2, 3....
Odd Consecutive Integers
Consecutive odd integers are the set of integers such that each integer in the set differs from the previous integer by a difference of 2 and each integer is an odd number.
Consecutive odd integers are odd integers that follow each other by the difference of 2. If x is an odd integer, then x + 2, x + 4, x + 6 and x + 8 are consecutive odd integers.
Examples:
1,3, 5, 7, 9, 11,…
-7, -5, -3, -1, 1,…
-25, -23, -21,….
You can represent consecutive odd integers with the following expression: 2n + 1 with n = 0, 1, 2, 3....
Consecutive Integers Formula
The consecutive integers formula is given by:
For odd consecutive integers
The general form of a consecutive odd integers formula is given by:
For even consecutive integers
The general form of a consecutive even integers formula is given by:
Where, “n” can be any integer.
Solved Examples: Consecutive Integers Problems
Example 1: John has a wooden board that is 5 feet long. He plans to make 4 shelves whose lengths are to be a series of consecutive even numbers. Find the length of each shelf in inches?
Solution:
Step 1: We know that consecutive even integers are even integers that follow each other by a difference of 2.
Let x = length of first shelf
x + 2 = length of second shelf
x + 4 = length of third shelf
x + 6 = length of fourth shelf
Step 2: Converting feet to inches
1 feet = 12 inches
Hence, 5 × 12 = 60 inches
Step 3: adding the 4 consecutive integers equal to 60
x + x + 2 + x + 4 + x + 6 = 60
Combine like terms
4x + 12 = 60
Isolate variable x
4x = 60 – 12
4x = 48
x = 12
Step 4: substitute the values of x
length of first shelf = x = 12 inches
length of second shelf = x + 2 = 14 inches
length of third shelf = x + 4 = 16 inches
length of fourth shelf = x + 6 = 18 inches
Therefore, the lengths of the shelves should be 12inch, 14 inch, 16 inch and 18 inch.
Let's check our answer
12 + 12 + 2 + 12 + 4 + 12 + 6 = 60
Example 2: If the sum of three consecutive integers is 81, Find the three integers and then find what is the product of the first and the third integer?
Solution:
Step 1: Let us assume the three consecutive integers: x, x + 1 and x + 2
Step 2 : Now, as given
x + x + 1 + x + 2 = 81
3x + 3 = 81
3x = 81 – 3
3x = 78
x = 78/3
x = 26
x + 1 = 27
x + 2 = 28
Product of the first and third integer = 26 × 28 = 728.
Quiz Time
Consecutive integers problems
The sum of two consecutive integers is 120. Find the value of the smaller integer.
The sum of two consecutive odd integers is 60. What are the integers?
Facts
The term consecutive numbers is used to frame word problems.
The sum of any two consecutive numbers is always odd. Example, 5 + 6 = 11
FAQs on Consecutive Integers
1. What are consecutive integers?
The numbers that follow each other are known as consecutive integers. They happen in a certain order or sequence. A set of natural numbers, for example, is a set of consecutive integers. Consecutive integers follow a series where each subsequent number is one more than the previous number, such that consecutive integers follow a sequence where each subsequent number is one more than the previous number. The mean and median of a set of consecutive integers (or numbers) are equal. x + 1 and x + 2 are two consecutive integers if x is an integer. In this article, we'll go through the definition of consecutive in math, as well as what consecutive numbers are, formulae, properties, and examples.
2. Write a note on the properties of consecutive integers?
Consecutive Integers have the following Properties:
Any two successive odd or even numbers have a difference of 2.
In any sequence of n consecutive integers, there will be exactly one number divisible by n. Any four integers in a row, for example, must be multiples of four; any 19 integers must be multiples of 19, and so on. Consider the following set of three consecutive integers: –1, 0, +1; there is no multiple of three here. When it comes to zero, this is a unique scenario. When an integer is multiplied by zero, the result is zero.
In a set of three successive integers, there may be two even numbers and one odd number, or two odd numbers and one even number, depending on the set that was started.
The total sum of m consecutive integers will be divisible by m if m is an odd number. For example, the sum of any five integers in a row is divisible by 5, and so on.
3. What is the formula of consecutive integers?
We can Deduce from the Above that the Successive Numbers are of the form:
x, x+1, x+2, x+3,...
where,
x is a positive integer, and
x + 1, x + 2, and so on are consecutive integers in order.
We suppose the first integer in a problem involving successive integers is x, and the succeeding integers may be derived by adding 1 to the preceding integer.
4. How consecutive integers are represented in algebraic representations
It is easy to recognize a set of consecutive integers in plain numbers.
But the algebraic representation of consecutive integers is different. The following are examples of representations of consecutive integers in algebraic form:
{n, n + 1, n + 2, n + 3, n + 4, n + 5, n+ 6}
{n - 4, n - 3, n - 2, n - 1, n, n + 1, n + 2, n + 3, n + 4}
{n + 11, n + 12, n + 13, n + 14, n + 15, n + 16, n+ 17}
For simplicity, let’s pick n = 5. The first set becomes the set of integers from 5 to 11 ; the second, from 1 to 8; and the third, from 16 to 22.
5. How Many Digits at the Maximum Count can a Set of Consecutive Integers have?
There is no limitation in the maximum count of digits in a set of consecutive integers. 1, 3, 5, 7... and 234, 456, 567, 789… both these sets are considered to be consecutive as they don’t have any gaps breaking the number sequence.