Consecutive numbers are numbers that come one after another in order, without skipping any number. They follow a natural sequence, starting from a smaller number and moving up step by step. Each number in the sequence is just one more than the previous number.

For example, 4, 5, 6, 7 are consecutive numbers because each number increases by 1. The difference between any two numbers next to each other is always 1.

These numbers are a basic part of math and are commonly used in counting, solving equations, and understanding patterns. Consecutive numbers are important in algebra and help us understand how numbers relate to each other in order.

One interesting fact about consecutive numbers is that the mean (average) and the median (middle value) of a group of consecutive numbers are always the same.

What are Consecutive Numbers?

Consecutive numbers are numbers that follow each other continuously. If x is any number then x and x + 1 are two numbers that are consecutive. For example, 14 and 15 as well as 21 and 22, are consecutive numbers. Consecutive numbers are those that follow each other in descending order from least to largest.

Examples of consecutive numbers are 1, 2, 3, 4, 5, 6, and so on are all consecutive numbers.

One of the numbers in a pair must be even while the other must be odd. The product of any number of consecutive numbers is always even because the product of an even number and an odd number is always even.

If n is a number, then n+1 and n+2 are the following two numbers.

Example of Consecutive Numbers: Consecutive numbers that follow each other in order:

1, 2, 3, 4, 5

-3, −2, −1, 0, 1, 2, 3, 4

6, 7, 8, 9, 10, 11, 12, 13

What are Consecutive Even Numbers?

If x is an even number, then x and x + 2 are consecutive even numbers.

Example: 8 and 10 are consecutive even numbers, as are 26 and 28.

Example of consecutive even numbers: If the sum of two consecutive even numbers is 194, Find the numbers.

Let the two numbers be x and x + 2.

\begin{matrix}
\therefore{x + x + 2}=194\\
2x+2=194\\
\text{Subtracting 2 from both sides}\\
2x+2-2=194-2\\
2x=192\\
\text{Dividing both sides by 2}\\
{2x\over{2}}={192\over{2}}\\
x=96
\end{matrix}

So the two consecutive numbers are 96 and 98.

Consecutive Even Numbers Formula

Sum of “n” natural even numbers = (n) (n + 1)

Sum of the square of “n” first or consecutive square even numbers:

\({2n(n+1)(2n+1)\over{3}}\)

What are Consecutive Odd Numbers?

If x is an odd number, then x and x + 2 are consecutive odd numbers.

Example: 7 and 9 are consecutive odd numbers, as are 31 and 33.

Example of consecutive odd numbers: If the sum of two consecutive odd numbers is 228, find the numbers.

Let the two consecutive odd numbers be x and x + 2.

\begin{matrix}
\therefore{x + x + 2}=228\\
2x+2=228\\
\text{Subtracting 2 from both sides}
2x+2-2=228-2\\
2x=226\\
\text{Dividing both sides by 2}\\
{2x\over{2}}={226\over{2}}\\
x=113
\end{matrix}

So the two consecutive numbers are 113 and 115.

Consecutive Odd Numbers Formula

Sum of “n” natural odd numbers = \(n^2\)

Sum of the square of “n” first or consecutive square even numbers:

\({n(4n^2-1)\over{3}}\)

Consecutive Numbers from 1 to 100: Consecutive numbers from 1 to 100 are all natural numbers between 1 to 100. That means the consecutive numbers are: 1, 2, 3, 4, 5, ….. 100.

Consecutive Natural Numbers: Two natural numbers that follow each other in order are called consecutive numbers. For example, 1 and 2 are two consecutive natural numbers.

HCF of Two Consecutive Numbers: When two consecutive numbers are added together, the HCF is always one. The reason for this is that other than 1, the two successive integers have no common element. As a result, 1 becomes the HCF of two consecutive numbers.

LCM of Two Consecutive Numbers: Now, let’s see the lcm of consecutive numbers. The product of any two consecutive numbers is the LCM of those values. For instance, because 2 and 3 are consecutive numbers, their LCM is 2 x 3 = 6. They have no common element because they are sequential numbers.

Sum of First Consecutive Numbers: The sum of the first n consecutive numbers = n/2[2a + (n – 1)d]. Now we will see the Sum of the First 100 Consecutive Numbers.

Sum of squares of first 100 consecutive numbers = 100/2[2 x 1 + (100 – 1)1]

= 50(2 + 99)

= 50 x 101

= 5050

Sum of Consecutive Numbers Formula

Let us assume the required sum = S

Therefore, \(S = 1 + 2 + 3 + 4 + 5 + …………. + n\)

It is an Arithmetic Progression whose first term = 1, last term = n and a number of terms = n. The difference between the two consecutive terms is 1.

The first term is \(a_1\), the second term is \(a_1 + d\), the third term is \(a_1 + 2d\), etc. This leads up to finding the sum of the arithmetic series, \(S_n\), by starting with the first term and successively adding the common difference.

Simplifying \(S_n = {n\over2}[2 a_1 + (n – 1)d]\)

We will find out the Sum of Squares of First n Consecutive Numbers using the above formula.

Put \(a_1\) = 1 and d = 1

Therefore, \(S_n = {n\over2}[2+ (n – 1)1]\) 

\(S_n = {n\over2}[2+ n – 1]\) 

\(S_n = {n\over2}[ n + 1]\) 

\(S_n = {n(n+1)\over2}\).

Types of Consecutive Numbers

Consecutive numbers are numbers that come one after another in order. There are different types of consecutive numbers, and we can group them into three main categories:

1. Consecutive Natural Numbers

• Natural numbers are the counting numbers we use every day like 1, 2, 3, etc.
• Consecutive natural numbers are numbers that follow each other without skipping.
• Example: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

2. Consecutive Even Numbers

• Even numbers are numbers that can be divided by 2.
• If even numbers follow one another with a gap of 2, they are called consecutive even numbers.
• Example: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20

3. Consecutive Odd Numbers

• Odd numbers are numbers that cannot be divided by 2 evenly.
• Consecutive odd numbers have a difference of 2 between each number.
• Example: 1, 3, 5, 7, 9, 11, 13, 15

4. Consecutive Integers

• Integers include both positive and negative whole numbers and 0.
• Consecutive integers are whole numbers that come right after each other on the number line.
• Positive consecutive integers: 0, 1, 2, 3, 4, 5...
• Negative consecutive integers: -1, -2, -3, -4, -5...

Properties of Consecutive Numbers

The properties of consecutive numbers are as follows:

• There will be precisely one number divisible by n in any sequence of n consecutive numbers. For example, any three numbers in a row must be multiples of three; any seventeen numbers must be multiples of seventeen, and so on. Now you might be wondering where the multiple of 3 is in the set –1, 0, +1, a collection of three consecutive numbers. This is a difficult situation. Because (any integer) times zero equals zero, zero turns out to be a multiple of every number.
• Depending on where we started, we could have two evens and one odd in a series of three consecutive numbers, or two odds and one even. Two evens and two odds are required in a collection of four consecutive numbers. In general, if we have an odd number of consecutive numbers, we may have more evens or odds, depending on the initial value; however, if we have an even number of consecutive numbers, the evens and odds must be evenly split.
• The sum of n consecutive numbers is divisible by n if n is an odd number. For example, the sum of any three numbers in a row is divisible by three; the total of any seven integers in a row is divisible by seven, and so on.

Learn about Arithmetic Mean

Consecutive Composite Numbers

Composite numbers are numbers that have more than two factors. In other words, they can be divided exactly by 1, themselves, and at least one more number. For example, 8, 9, and 10 are composite numbers.

When we arrange composite numbers one after the other, they are called consecutive composite numbers. These numbers appear in order without any gaps. Examples include:

• 8, 9, 10
• 14, 15, 16
• 24, 25, 26, 27, 28

All the numbers in each group are composite and listed one after the other.

Tips to Remember

• To find missing numbers in a sequence, write them in increasing order and look at the difference between each pair.
• For a pattern of odd consecutive numbers, use the formula:
  2n + 1 (where n is any whole number)
• For even consecutive numbers, use the formula:
  2n (where n is any whole number)

These tips can help you easily identify and work with consecutive composite numbers.

Solved Examples of Consecutive Numbers

Let’s learn some examples of consecutive numbers that come in exams.

Example 1: The sum of three consecutive integers is 8484. Find the three consecutive integers.

Solution: Let the three consecutive integers n, n + 1, n + 2.

Let the three consecutive integers n, n + 1, n + 2.
\begin{matrix}
\therefore{n + n + 1 + n + 2} = 84\\
3n+3=84\\
\text{Subtracting 3 from both sides}
3n+3-3=84-3\\
3n=81\\
\text{Dividing both sides by 3}\\
{3n\over{3}}={81\over{3}}\\
x=27
\end{matrix}

Example 2: Find four consecutive integers whose sum is 238.

Solution: Let the four consecutive integers n, n + 1, n + 2, n + 3.

\begin{matrix}
\therefore{n + n + 1 + n + 2 + n + 3} = 238\\
4n+6=238\\
\text{Subtracting 6 from both sides}
4n+6-6=238-6\\
4n=232\\
\text{Dividing both sides by 4}\\
{4n\over{4}}={232\over{4}}\\
x=58
\end{matrix}

Example 3: The sum of five consecutive integers is 145. Find the five consecutive integers.

Solution:

Let the five consecutive integers be:
n, n + 1, n + 2, n + 3, and n + 4

Now add them:

n + (n + 1) + (n + 2) + (n + 3) + (n + 4)
= 5n + 10

We are given:
5n + 10 = 145

Subtract 10 from both sides:
5n = 135

Divide both sides by 5:
n = 27

So, the five consecutive integers are: 27, 28, 29, 30, and 31

Check:
27 + 28 + 29 + 30 + 31 = 145.

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