Consecutive Integers are the integers that follow each other, i.e. while continuously writing integers they come next to each other. they have a difference of one(1). For example, …-3, -2, -1, 0, 1, 2, 3,… this is a sequence of consecutive integers. Apart from that natural numbers are also called consecutive integers because they are all integers and the difference between two consecutive natural numbers is always 1.
Thus, we can say that consecutive integers are the number that follows a regular pattern of writing and there is a fixed difference between any two consecutive integers, i.e. they have a difference of One(1). We represent any two consecutive integers as n, n + 1 where n ϵ Z.
In this article, we will learn about Consecutive Integers, Consecutive Even Integers, Consecutive Odd Integers, Examples, and others in detail.
Before learning in detail about Consecutive Integers let’s first understand what is the meaning of Consecutive in maths.
Consecutive Meaning
Consecutive in Mathematics is defined as the number that comes next to each other. Suppose we have to write consecutive natural numbers then they are 1, 2, 3,…, suppose we have to write consecutive whole number then they are, 0, 1, 2, 3,… Now one thing to note is that their are no consecutive rational numbers or consecutive irrational numbers, because between any two rational numbers and irrational numbers there are infinite rational and irrational number.
What are Consecutive Integers?
Consecutive integers are the integers in mathematics that follow each other. They have a difference of 1 and they follow a regular pattern. The consecutive integers are always arranged in ascending order and written from smallest to largest.
Sequence of consecutive integer is represented as, n, n + 1, n + 2, n + 3, …
General Term of Consecutive Integer = n where n ϵ Z
Examples of Consecutive Integers
Some examples of consecutive integers include,
- 0, 1, 2, 3, 4, 5, …
- -5, -4, -3, -2, -1, 0, …
- -2, -1, 0, 1, 2,…
- 100, 101, 102, 103, …
From any of the example we can see that the difference between two consecutive integers is always one.
- 101 – 100 = 1
- (-3) – (-2) = 1
- 4 – 3 = 1
Consecutive Even Integers
We know that even integers are multiples of 2. So, if we list the set of even numbers in ascending order, they may be expressed as …,-4, -2, 0, 2, 4, 6, 8, 10, …
Thus, they are even consecutive number. We can see that the difference between each consecutive even integer is 2. The even consecutive number are represented in general form as, 2n, 2n + 2, 2n + 4, … where n ϵ Z
For example,
- …, 0, 2, 4, 6, …
- …, -4, -2, 0, 2, 4, ….
Consecutive Odd Integers
We know that odd integers that on dividing by 2 gives one(1) as remainder. So, if we list the set of odd numbers in ascending order, they may be expressed as …,-3, -1, 0, 1, …
Thus, they are odd consecutive number. We can see that the difference between each consecutive odd integer is 2. The odd consecutive number are represented in general form as, 2n + 1, 2n + 3, 2n + 5, … where n ϵ Z
For example,
- …, 1, 3, 5, 7, …
- …, -3, -1, 0, 1, 3, ….
The formula to calculate the consecutive integers is very basic and uses the concept of basic algebraic expressions.
For any given integer n the formula for the consecutive integer is n + 1
For Even Consecutive Integers
- The formula to calculate the even consecutive odd integer is 2n
For Odd Consecutive Integers
- The formula to calculate the odd consecutive odd integer is 2n +1
Consecutive Positive Integers
The consecutive positive integers are the consecutive numbers that are positive in nature, i,e, they are the positive natural numbers, for example 1, 2, 3, 4, 5,… are consecutive positive numbers. The difference between two consecutive integres is constant and is always equal to 1.
Three Consecutive Integers
Product of three consecutive integers (apart from 0) is always divisible by 6 this is shown as, suppose we have three consecutive integers then they are represented as,
n, n + 1, n + 2
Now the product of these three integers is,
(n)(n +1)(n + 2) = n3 + 3n2 + 2n
The above expression is always divisible by 6 and can be proved using Mathematical Induction. Thus, we can say that product of three consecutive integers is always divisible by 6.
For example,
- 1 × 2 × 3 = 6 {Divisible by 6}
- 5 × 6 × 7 = 420 {Divisible by 6}
- 11 × 12 × 13 = 1716 {Divisible by 6}, etc
Properties of Consecutive Integers
Various properties of the consecutive integers are,
- Difference between two consecutive integers is always constant and is equal to 1.
- Difference between two consecutive even integers is always constant and is equal to 2.
- Difference between two consecutive odd integers is always constant and is equal to 2.
- Product of three consecutive integers (apart from 0) is always divisible by 6.
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Consecutive Integers Solved Examples
Example 1: Find three consecutive integers after 64.
Solution:
Three consecutive integers sequence after x will be x + 1, x + 2, and x + 3
So, three consecutive integers after 64 are,
= 64 + 1, 64 + 2, and 64 + 3
= 65, 66, and 67
Therefore, three consecutive integer after 64 will be 65, 66, and 67
Example 2: Find 5 consecutive even integers of -10?
Solution:
Consecutive Integer after – 10,
Therefore, Let – 10 be x
So the next five consecutive even integers will be
= x + 2, x + 4, x + 6, x + 8, and x + 10
= (-10 + 2), (-10 + 4), (-10 + 6), (-10 +8), and ( -10 +10)
= -8, -6, -4, -2, and 0
Therefore, the five even consecutive integers will be have -10, -8, -6, -4, -2, 0.
Example 3: Find the smallest number if the sum of four consecutive odd integers is 288?
Solution:
Sum of four consecutive odd integers are 288
So, four consecutive odd integers sequence will be x, x + 2, x + 4 and x + 6
According to question,
x + x + 2 + x + 4 + x + 6 = 288
4x + 12 = 288
4x = 288 – 12
4x = 276
x = 276/4
x = 69
Hence, the smallest number will be 69.
Example 4: If the sum of four consecutive integers is 302, then what is the product of the first and the third integer?
Solution:
Lets assume four consecutive integers are x, x + 1, x + 2, x + 3
Sum of four consecutive integers are 302
Therefore,
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
Product of the first and third integer = 74 × 76 = 5624
Example 5: Find two consecutive positive integers sum of whose squares is 265?
Solution:
Two consecutive positive integers sum of whose squares is 265
Let’s assume one integer be x, then the other integer is x + 1,
So,
x2 + (x + 1)2 = 265
x2 + x2 + 1 + 2x = 265 [Algebraic Identity (a + b)2 = a2 + 2ab + b2]
2x2 + 2x + 1 = 265
2x2 + 2x + 1 – 265 = 0
2x2 + 2x – 264 = 0
x2 + x – 132 = 0
x2 + 12x – 11x – 132 = 0
x(x + 12) – 11 (x + 12) = 0
(x +12) (x -11) = 0
x = -12 or x = 11
Since, we need a positive integer,
Therefore x = -12 is rejected.
Thus, x = 11
x + 1 = 11 + 1 = 12
Hence first two positive consecutive integers are 11, 12.
Practice Questions on Consecutive Integers
Q1: Find the next 10 consecutive integers from -20
Q2: Find the sum of first 5 consecutive even numbers
Q3: Find the sum of first 7 consecutive even numbers
Q4: Find the product of three consecutive integers 9, 10 and 11
FAQS on Consecutive Integers
1. What are Consecutive Integers?
Consecutive Integers are the integers that are arranged in such a manner that the difference between any two successive number is always 1. For example, 2, 3, 4, 5,… are the consecutive integers.
2. What are Even Consecutive Integers?
Consecutive even numbers are the consecutive numbers that are even in nature are called consecutive even numbers. The difference between two consecutive even number is always 2. Even numbers are the numbers that are divisible by 2. Example of even consecutive integers is, 4, 6, 8, 10,…
3. What are Odd Consecutive Integers?
Consecutive odd numbers are the consecutive numbers that are odd in nature are called consecutive odd numbers. The difference between two consecutive odd number is always 2. Odd numbers are the numbers that are not divisible by 2 and they always leaves 1 as remainder. Example of odd consecutive integers is, 3, 5, 7, 9, 11,…
4. What is Consecutive Integers Formula?
Consecutive integer formula is the formula that is used to find the consecutive integers. Suppose we have an integer n then the consecutive integers is n + 1.
5. Can Consecutive Integers be Negative?
Yes, consecutive integers can be negative. For example …,-3, -2, -1, 0, 1, 2, 3,… are negative integers that are consecutive.
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