Consecutive Integers Formula: Definition, Roles, Properties

Consecutive integers are the numbers that follow each other. They follow each other in a sequence or in order. For example, a set/group of natural numbers are consecutive integers. Consecutive Maths represents an unbroken sequence or following continuously so that consecutive integers follow a sequence where each subsequent number is one more than the previous number. Integers are essential in our regular life besides maths paper. Integers are used to calculate the efficiency of any positive and negative numbers in all fields. The consecutive integers follow each other from the smallest number to the most significant digit.

Also Read: Meaning of Consecutive Numbers

Key terms: Consecutive Integer, Consecutive Integer Formula, Even Consecutive Integer, Odd Consecutive Integer.

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Consecutive integer

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Consecutive integers mean a set of natural numbers that follow each other. They follow a sequence and are in order. The Consecutive term, meaning in maths, is an unbroken sequence or following so that consecutive integers are those when each subsequent number is one more than the previous number.

Role of consecutive integers in maths

In mathematics, the consecutive integers formula plays a part in solving maths problems. If a sum and difference are given in the question and asked to find out the two or more absent consecutive integers, this formula helps you. The consecutive integers follow each other in a fixed sequence, where we can use this formula to check whether these digits are in a row or not.

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Types of Consecutive integers:

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1. Even consecutive integers (2,4,6,8,...)
2. Odd consecutive integers(3,5,7,9,...)
3. Normal Consecutive integers (2,3,4,5…)

The consecutive integers may be positive or negative, but both sets follow an increasing pattern. For understanding this, an example of both types of consecutive integers are as follows:

Example: 

• Positive integers: 1,2,3,4,5,...

In this example, you can express this n+1, with n = 0,1,2,... by adding 1 to 0

• Negative integers: -1,-2, -3, -4, -5…. 

We can express this set as 1-n, with n= 2,3,4,5 by subtracting from 0.

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Consecutive integers

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Consecutive integers are the digits that follow a fixed and increasing sequence, and each integer is more than the previous digit. In a practical tone, we can represent consecutive integers as n,n+1, n+2, n+3.

 where `n’ represents an integer.

Example: 12,13,14,15

Even consecutive Integers

Consecutive even integers are different from their previous integer having a difference of 2 and divisible by 2. Consecutive even integers are those digits that follow each other with a difference of 2. We can represent an even consecutive integer as 2n. Here 2n is an even integer, (2n+2) and (2n+4). You can find out the two even integers with the help of this formula.

For example: 

Ques. If 2n is 8, an even integer, find out its consecutive integers?

Ans. 

If 2n=8

Formula: 2n, 2n+2, 2n+4…..

(8+2) and (8+4)=10 and 12

So the consecutive even integers are 10 and 12.

As per the above example, the difference between two even consecutive integers is 2, but sometimes you can get a consecutive integer but not even. Example: 

2,4,6,8,10….

-8,-6,-4,-2,0…..

64,66,68,70….

At last we can represent the consecutive even integers as 2n+2 where n= 0,1,2,3,...

Consecutive Odd Integers

Consecutive odd integers represent the set of integers where each integer has a difference of 2 from the previous integers, and these are odd integers. In other words, consecutive odd integers follow each other with a difference of 2. For example: if we put n as an odd integer, then the consecutive odd integers are n+2, n+4, n+6.

The value of n :

• 1,3,5,7,9…..
• -7,-5,-3,-1,0…..
• -77, -75, -73….

To find the next consecutive integers we can use the formula: 2n+1,here n= 0,1,2,3…

For example: if 2n+1=9, find out the next odd integer?

Sol.: if 2n+1=9

Formula :2n+1, 2n+3, 2n+5….

(9+2),(9+4)=11,13

The next two integers are 11,13.

There is a difference of 2 between two integers in consecutive odd integers, but there are successive integers but not odd in some cases. If n is an odd number, then the total sum of consecutive integers will be divisible with n. 

For example, if there are five integers in a row, the total sum of integers is divisible by 5.

Consecutive integers formula

The consecutive integer: n+1

For even consecutive integers: 2n

For odd consecutive integers: 2n+1

Properties: 

• We can represent consecutive integers in plain numbers and algebraic patterns also but both patterns are slightly different. For example:

(n,n+1, n+2, n+3, n+4,n+5)

(n-2, n-1,n, n+1, n+2, n+3)

(n+21, n+22, n+23, n+24)

Here, we assume that n=3. 

The first integers set from 3 to 8; the second one, from 1 to 6, and the third, from 24 to 27.

 If there is no gap between the sequence of two integers, then there is no maximum count in a set of consecutive integers. such as 2, 4, 6, 8

 123,345,567….

We can use the consecutive number to seek out word problems. And the sum of two consecutive numbers is always odd. Such as 11+12=23.

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Things to Remember:

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• In a set of consecutive integers, the mean and median are equal.
• The difference between any two consecutive odd or even integers is 2.
• 0 is neither positive nor negative.
• ½ and 0.5 are not defined as integers because they are not whole numbers.

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Sample Questions

Ques. in a quiz, positive marks are for correct answers and negative marks for incorrect answers. Suppose Jack scores in five successive rounds were 25, -5-10,15, and 10. What was his total at the end? ( 2 Marks)

Ques. At Srinagar the temperature was -5° C on Monday, and then it dropped by two °C on Tuesday. What was the temperature of Srinagar on Tuesday? On Wednesday, it rose by 4°C. What was the temperature on this day?( 3 Marks)

Ques. Write down a pair of integers whose: ( 3 Marks)

• Sum is -7

• Difference is -10

• Sum is 0

Ques. A plane is flying at the height of 5000m above sea level. At a particular point, it is exactly above a submarine floating 1200m below sea level. What is the vertical distance between them? ( 3 Marks)

Ques. John deposits $ 2000 in his bank account and withdraws $1642 from it, the next day. If the withdrawal of the amount from the account is represented by a negative integer, then how will you represent the amount deposited? Find the balance in John’s accounts after the withdrawal? ( 2 Marks)

Ques. An elevator descends into a mine shaft at the rate of 6m/min. If the descent starts from 10 above the ground level, how long will it take to reach - 350 m? ( 3 Marks)

Ques. In a quiz, team A scored -40,10,0and team B scored 10,0,-40 in three successive rounds. Which team scored more? Can we say that we can add integers in any order?( 2 Marks)

Ques. If the sum of four consecutive odd integers is 184. Find the smallest number. ( 2 Marks)

Ques. Find three consecutive integers that add up to 51. ( 3 Marks)

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