Consecutive Integers: Definition & Formula

Consecutive integers are numbers that progress in a patterned order. Explore the formulas used to establish consecutive integers, and learn how both even and odd integers work as consecutive integers through examples

What is a Consecutive Integer?

An integer is simply a number like 0, 1, 2, 3, and 4, but unlike whole numbers, integers also include negative numbers like -1, -2, -3 and -4. An integer cannot be a decimal or a fraction.

Consecutive integers are simply integers that follow each other in a patterned order, usually just one number after the other, like 1, 2, 3 and 4!

Examples of Consecutive Integers

Example 1:

Let’s imagine we are counting down to the new year!

The countdown to 1 when we are about to welcome a new year is an example of consecutive integers. They follow each other in order, just backwards.

Example 2:

Let’s imagine a teacher is counting the heads of students as they load the bus for a field trip.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12!

The teacher has just counted all 12 children. Numbers 1-12 represent consecutive integers!

Negative Consecutive Integers

We cannot forget that consecutive integers also include negative numbers.

Take a look at this number line indicating consecutive numbers from -3 to 3:

Even and Odd Consecutive Integers

Even consecutive integers are even numbers that follow each other in order.

The easiest example would be 2, 4, 6, 8 and 10.

246, 248, 250 and 252 is another example of even consecutive integers.

Odd consecutive integers are odd numbers that follow

each other in order.

The easiest example would be 1, 3, 5, 7 and 9.

157, 159, 161 and 163 is another example of odd consecutive integers.

Examples of Use of Even and Odd Consecutive Integers

Example 1 (even):

Ted, a middle school coach, wants to make two fair teams from his physical education class to play an indoor soccer game. He asks the class to line up and taps every other person, saying ‘2, 4, 6, 8’ etc. until the 20th person to make team A. He announces that the students that he did not tap (the students in odd position) will make up team B.

So, Ted has used even consecutive integers (2, 4, 6, 8, 10, 12, 14, 16, 18 and 20) to make teams for the soccer game.

Example 2 (odd):

Mira has 30 gift bags for her 10th birthday pool party. She has 15 boys and 15 girls coming to her party. Her mother has lined up the bags so that they alternate between boy and girl. Mira’s mother has already filled the girl bags with goodies. She asks Mira for help in filling the boy bags. Mira’s mother asks her to only fill the bags from the very first bag (#1) to the second to last bag (#29) using odd consecutive integers.

Mira then fills bags 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 and 29 for the boy gift bags.

So, Mira has used odd consecutive integers to fill up the party bags.

Formula for Consecutive Integers

The formula for consecutive integers is pretty straightforward.

If x is the first consecutive integer, then x+1 will be the second, x+2 will be the third, x+3 will be the fourth, and so on.

Consecutive Integers Formula

In mathematics, you often come across problem statements wherein you are required to find two or more consecutive integers if their sum or difference is given to you. A number of times the clause of these consecutive numbers being odd or even is also added. Before simplifying the problem statement and forming its equation, you are supposed to use a variable for one integer and then represent it in the form of its consecutive integers. Let me explain this with the help of an example.

Say you have to find two consecutive integers whose sum is 89. How do you go about this problem? You first take a variable, say x, the value of which is unknown to you. Next, you are supposed to take another number. Since the problem requires you to use two consecutive integers, the integer next to x will be (x + 1). Now as per the problem, the sum of x and (x+1) is 89. We represent this in the form of an equation: x + (x + 1) = 89. Solving this equation we get x as 44 and the next integer (x + 1) as 45, the sum of which is 89.

Similarly, knowing the consecutive integer formula finds application in a number of mathematical problems. In this section, we will go through some of these formulas.

Consecutive Integer Formula

If n is an integer, (n + 1) and (n + 2) will be the next two consecutive integers. For example, let n be 1. We find its consecutive integers as (1 + 1) and (1 + 2), or 2 and 3.

Hence, the formula:

n, n+1, n+2, n+3,…

Even Consecutive Integer Formula

In mathematics, we represent an even integer as 2n. If 2n is an even integer, (2n + 2) and (2n + 4) will be the next two even consecutive integers. For example, let 2n be 4, which is an even integer. We find its consecutive integers as (4 + 2) and (4 + 4), or 6 and 8.

Hence, the formula:

2n, 2n+2, 2n+4, 2n+6,…

Note that the difference between two even consecutive integers here is 2, otherwise, we would end up with an integer which is consecutive but not even.

Odd Consecutive Integer Formula

In mathematics, we represent an odd integer as 2n + 1. If 2n + 1 is an odd integer, (2n + 3) and (2n + 5) will be the next two odd consecutive integers. For example, let 2n + 1 be 7, which is an odd integer. We find its consecutive integers as (7 + 2) and (7 + 4), or 9 and 11.

Hence, the formula:

2n+1, 2n+3, 2n+5, 2n+7,…

Note that the difference between two odd consecutive integers here is 2, otherwise, we would end up with an integer which is consecutive but not odd.

Solved Examples

Question: Find three consecutive integers of 76.

Solution:

Let 76 be n. So the next three consecutive integers will be n + 1, n + 2 and n + 3.

76 + 1, 76 + 2 and 76 + 3 or 77, 78 and 79

Therefore, we have 76, 77, 78, 79

Question: Find three consecutive even integers of -8.

Solution:

Let -8 be 2n. So the next three consecutive integers will be 2n + 2, 2n + 4 and 2n + 6.

-8 + 2, -8 + 4 and -8 + 6 or -6, -4 and -2

Therefore, we have -8, -6, -4, -2

Examples of Consecutive Integers

• 0, 1, 2, 3, 4, …
• -3, -2, -1, 0, 1, ….
• -12, -11, -10, …

We can see that each integer in any list of consecutive integers is obtained by adding 1 to its previous integer. Thus, two consecutive integers differ by 1.

Let us understand the consecutive integers formula using solved examples in the following sections.

What is Consecutive Integers Formula?

Using the above explanation, we conclude that the consecutive integers are of the 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 numbers (or) two consecutive odd numbers differ by 2. So any two consecutive even numbers (or) consecutive odd numbers are of the form:

x, x + 2, x + 4, …

where,

x is an integer, and

x + 2, x + 4, .. are successive even/odd consecutive integers in sequence.

Examples Using Consecutive Integers Formula

Example 1: Find the set of three consecutive integers whose sum is 78.

Solution:

To find: Set of three consecutive integers whose sum is 78.

Using the formula of consecutive integers, we can assume the three consecutive integers to be 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

Subtracting 3 from both sides,

3x = 75

Dividing both sides by 3,

x = 25

So the three consecutive integers are:

x = 25

x + 1 = 25 + 1 = 26

x + 2 = 25 + 2 = 27

Answer: The required integers are 25, 26, and 27.

Example 2: Find three consecutive odd integers following the number -11.

Solution:

To find: Three consecutive odd integers following number -11.

We know that the consecutive odd integers differ by 2 and are of form x, x + 2, x + 4, …

Let us assume that x = -11.

Then three consecutive odd integers of x are

x + 2 = -11 + 2 = -9

x + 4 = -11 + 4 = -7

x + 6 = -11 + 6 = -5

Answer: The three consecutive integers of -11 are -9, -7, and -5.

Example 1: 7, 8, 9, 10, 11

Here we see a set of consecutive integers that are found by adding one to the next number, which is represented as the expression a+1, with a = 0, 1, 2, ….

Example 2: -3, -4, -5, -6, -7

Here we see a set of consecutive integers that is found by subtracting 1 from the next number, which is represented as the expression 1 – a, with a = 2, 3, 4, ….

Consecutive Numbers

Numbers that follow each other in a regular counting order or pattern are called consecutive numbers. They are written in a series where the difference between the numbers is fixed and where no numbers are skipped in between. Let us learn more about consecutive numbers in this article.

What are Consecutive Numbers?

To understand consecutive numbers, we first need to understand the concept of predecessors and successors. The number that is written immediately before a number is called its predecessor. The number that is written immediately after a number is called its successor. For example, consider the list of natural numbers,1, 2, 3, 4, and 5. The predecessor of 2 is 1, and the successor of 2 is 3. Consecutive numbers are numbers that follow each other in order from the smallest number to the largest number. They usually have a difference of 1 between every two numbers. Note that the difference between any predecessor-successor pair is fixed. Let us look at a few examples of consecutive numbers.

In the above example, the difference between any predecessor-successor pair is 1. If we denote the 1st number as n, then the consecutive numbers in the series will be n, n+1, n+2, n+3, n+4, and so on.

Consecutive Even Numbers

We know that even numbers are those numbers that end in 0, 2, 4, 6, and 8. Now, let us consider the set of even numbers from 2 to 12 and write them in ascending order. The numbers are arranged as 2, 4, 6, 8, 10, 12 when written from the smallest to the largest. We can see that the difference between any predecessor-successor pair is 2. Therefore, these numbers form the list of consecutive even numbers.

Consecutive Odd Numbers

We know that odd numbers are those numbers which cannot be completely divided by 2. When we arrange them in ascending order, we can see that the difference between them is always 2. For example, the numbers 3, 5, 7, 9, and 11 are called consecutive odd numbers because the difference between any predecessor-successor pair is 2, like, 5 – 3 = 2 and 7 – 5 = 2.

Properties of Consecutive Numbers

Consecutive numbers are numbers that follow each other in order from the smallest number to the largest number. The following points show the properties of consecutive numbers.

• In consecutive numbers, the difference between any predecessor-successor pair is fixed. If we denote the 1st number as n, then the consecutive numbers in the series will be n, n+1, n+2, n+3, n+4, and so on.
• For any two consecutive odd numbers, the difference is 2. For example, 3 and 5 are two consecutive odd numbers, their difference = 5 – 3 = 2.
• For any two consecutive even numbers, the difference is 2. For example, 6 and 8 are two consecutive even numbers, their difference = 8 – 6 = 2.
• If ‘n’ is an odd number, then the sum of ‘n’ consecutive numbers will be divisible by ‘n’. For example, the sum of these 3 consecutive numbers is 5+6+7=18 and 18 is divisible by 3.

Consecutive Numbers Formula

For a number n, the next two consecutive numbers are (n + 1) and (n + 2). Given below are more consecutive number formulas.

• The formula for adding ‘n’ consecutive numbers = [a + (a + 1) + (a + 2) + …. {a + (n-1)}]. So, the sum of ‘n’ consecutive numbers or sum of ‘n’ terms of AP (Arithmetic Progression) = (n/2) × (first number + last number).
• Even Consecutive Numbers Formula = 2n, 2n+2, 2n+4, 2n+6,…
• Odd Consecutive Numbers Formula = 2n+1, 2n+3, 2n+5, 2n+7,…

Important Points

Here is a list of some important points that should be remembered while working with consecutive numbers.

• To find the missing numbers in a series, write the numbers in ascending order and find the difference between any predecessor-successor pair.
• If we denote the 1^st number as n, then the consecutive numbers in the series will be n+1, n+2, n+3, n+4, and so on.
• If we denote the 1^st integer as n, the consecutive even or consecutive odd integers will be n+2, n+4, n+6, n+8, and so on.
• For an odd consecutive number, the general formula = 2n+1 (where ‘n’ = any integer)
• For an even consecutive number, the general formula = 2n (where ‘n’ = any integer)

Consecutive Numbers Examples

FAQs on Consecutive Numbers

What are Consecutive Numbers in Math?

Consecutive numbers are numbers that follow each other in order from the smallest number to the largest number. The difference between consecutive numbers is always fixed and it follows a pattern. For example 1, 2, 3 are the first three consecutive natural numbers.

What are Consecutive Positive Numbers?

Consecutive positive numbers are the set of positive numbers whose difference is 1. For example, 1, 2, 3, 4, 5, 6 … is the set of consecutive positive numbers.

Can Consecutive Numbers be Decimals?

No, consecutive numbers cannot be decimal numbers because there are several numbers between every two decimal numbers. For example, in the list of these numbers: 3.1, 3.2 and 3.3…, many decimal numbers like 3.11, 3.111, 3.1111… exist between them.

When are 2 Numbers Considered Consecutive Numbers?

Two numbers that follow each other in order are called two consecutive numbers. For example:

• 1 and 2 are two consecutive natural numbers.
• 3 and 6 are two consecutive multiples of 3.
• 10, 20 are two consecutive multiples of 10.

What are Odd Consecutive Numbers?

We know that odd numbers are those numbers which are not completely divisible by 2. When we arrange odd numbers in ascending order, we can see that the difference between them is always 2. Therefore, when odd numbers are arranged as 3, 5, 7, 9, and 11, they are called consecutive odd numbers because these numbers have a fixed difference of 2 between any predecessor-successor pair.

What are Non Consecutive Numbers?

Non consecutive numbers are those numbers that form a list in which there is no pattern or any fixed difference between a predecessor and a successor. For example, 2, 5, 17, 21 and so on are non consecutive numbers.

Can Consecutive Numbers be Fractions?

No, just like decimals, fractions cannot be considered as consecutive numbers because there are several fractions existing in between two fractions.

1. What are the Consecutive Integers Properties

Answer : The following are the consecutive integers properties numbers:

• The difference between any two consecutive either odd or even integers is 2.
• Any set of integers has accurately one number divisible by n. For example, any six integers in a row must have a multiple of 6; any 15 integers will have one multiple of 15 and so on.
•  Consider a set of three consecutive integers: {–1, 0, +1}, here we observe multiple of 3 does not exist. This is the special case when it turns out to zero. 
• Depending upon the set which has been started, there might be two even numbers and one odd number, or two odd numbers and one even number in a set of 3 consecutive integers. In a set of 4 consecutive integers,it has two even and two odd numbers. Depending on the starting value, if a set has an odd number of consecutive integers, there will be a chance of more evens or more odds. But if a set has an even number of consecutive integers, the even and odd integers will be in equal number.
• If x is an odd number, then the total sum of x consecutive integers will be divisible by x. For example, for any three integers in a row, the sum is divisible by 3, etc.,

2. How consecutive integers are represented in algebraic representations

Answer : 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.

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. 

1. How Many Digits can a Set of Consecutive Integers have in Maximum?

Ans: In a set of consecutive integers, there is no limit in the maximum count of the digits. If we consider two sets, 1, 2, 3, …… and 996, 997, 988, 999, ….. then both these sets are termed as consecutive integers because they do not have any gaps which will break the number sequence.

2. If the Sum of Three Consecutive Integers is 99, Find out the Three Integers. Also, Find out the Product of the First and Third Integer.

Ans: Let say, that the three consecutive integers: x, x + 1, x + 2

According to the sum of consecutive integers formula,

x + x + 1 + x + 2 = 99

3x + 3 = 99

3x = 99 – 3

3x = 96

x = 96 ÷ 3

x = 32

x + 1 = 33

x + 2 = 34

The product of first and third integer = 33 × 34 = 1088

3. Explain with an Example What is Consecutive Even Integers Formula and Consecutive Odd Integers Formula.

Ans:

Consecutive Even Integers Formula: Let 2 is an integer, then applying consecutive even integers formula, the set of even integers are:

n, n + 2, n + 4, n + 6, …

= 2, 2 + 2, 2 + 4, 2 + 6, ….

= 2, 4, 6, 8, ….

Consecutive Even Integers Formula: Let 3 is an integer then applying consecutive odd integers formula the set of odd integers are:

n, n + 2, n + 4, n + 6, …

= 3, 3 + 2, 3 + 4, 3 + 6, ….

= 3, 5, 7, 9, ….

Introduction

When you start counting natural numbers what are you doing 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 any integer.

For example: 23, 24, 25

Look at the following two sets. 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 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

n+1

For odd consecutive integers:

The general form of a consecutive odd integers formula is given by

2n+1

For even consecutive integers:

The general form of a consecutive even integers formula is given by

2n

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 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, 14inch, 16inch and 18inch.

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

1. The sum of two consecutive integers is 120. Find the value of the smaller integer.
2. 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

Example Questions

Example Question #1 : Consecutive Integers

Four consecutive odd integers have a sum of 32. What are the integers?Possible Answers:

6,8,10,12

7,8,10,12

3,5,7,9

7,8,9,10

5,7,9,11Correct answer:

5,7,9,11Explanation:

Consecutive odd integers can be represented as x, x+2, x+4, and x+6.

We know that the sum of these integers is 32. We can add the terms together and set it equal to 32:

x + (x+2) + (x+4) + (x+6) = 32

4x + 12 = 32

4x = 20

x = 5; x+2=7; x+4 = 9; x+6 = 11

Our integers are 5, 7, 9, and 11.

Example Question #1 : Consecutive Integers

What is the sum of all of the four-digit integers that can be created with the digits 1, 2, 3, and 4?Possible Answers:

482,912

37,891

711,040

5994

48,758Correct answer:

711,040Explanation:

First we need to find out how many possible numbers there are. The number of possible four-digit numbers with four different digits is simply 4 * 4 * 4 * 4 = 256.  

To find the sum, the formula we must remember is sum = average * number of values. The last piece that’s missing in this formula is the average. To find this, we can average the first and last number, since the numbers are consecutive. The smallest number that can be created from 1, 2, 3, and 4 is 1111, and the largest number possible is 4444. Then the average is (1111 + 4444)/2.

So sum = 5555/2 * 256 = 711,040.

Example Question #1 : Consecutive Integers

Quantity A: The sum of all integers from 49 to 98 inclusive.

Quantity B: The sum of all integers from 51 to 99 inclusive.Possible Answers:

The relationship cannot be determined from the information given.

Quantity B is greater.

The two quantities are equal.

Quantity A is greater.Correct answer:

The two quantities are equal.Explanation:

For each quantity, only count the integers that aren’t in the other quantity. Both quantities include the numbers 51 to 98, so those numbers won’t affect which is greater. Only Quantity A has 49 and 50 (for a total of 99) and only Quantity B has 99. Since the excluded numbers from both quantities equal 99, you can conclude that the 2 quantities are equal.

Example Question #3 : Consecutive Integers

What is the sum of all the integers between 1 and 69, inclusive?Possible Answers:

2298

2509

2211

2415

2312

Correct answer:

2415

Explanation:

The formula here is sum = average value * number of values.  Since this is a consecutive series, the average can be found by averaging only the first and last terms: (1 + 69)/2 = 35.  

sum = average * number of values = 35 * 69 = 2415

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