## **Table of Contents**

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**Introduction to Consecutive Numbers**

To understand consecutive numbers, we will 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**.

**What are Consecutive Numbers?**

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: The difference between any predecessor-successor pair is fixed.

Let’s look at a few examples of consecutive numbers.

**Example 1:**

In the above example, the difference between any predecessor-successor pair is 1

If we denote the 1^{st }number as \(n\), then the consicutive numbers in the series will be \(n+1\), \(n+2\), \(n+3\), \(n+4\), and so on.

Here is another example for you.

The difference between any predecessor-successor pair in this example is always 6, as they are all multiples of 6

Do you think there will be an odd number in the list of consecutive multiples of 6?

**Consecutive Even Numbers**

We know that even numbers are those numbers which 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 ordered 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

Can you identify the consecutive even numbers between 20 and 30?

**Consecutive Odd Numbers**

We know that odd numbers are one less or one more than the even numbers.

When we arrange them in ascending order, we can see that the difference between them is always 2

In the above example, the difference between any predecessor-successor pair is 2

\(5-3 = 2 \)

\(7-5 = 2 \) and so on.

**Consecutive Even and Odd Integers**

We know that integers are positive and negative numbers including zero. ** **

**Consecutive Even Integers **

We know that even integers end in \(2,4,6,8,0\) and they include negative numbers.

Let us write the consecutive even integers between \(-6\) and \(6\)

When we arrange these numbers in order from the least value to the greatest value, we get \( -6,-4, -2, 0, 2, 4, 6 \)

Note that the difference between the successive predecessor and successor is 2

**Consecutive Odd Integers**

We know that odd integers end in \(1,3,5,7,9\) and they include negative numbers.

Let us write the consecutive odd integers between \(-5) and \(5)

When we arrange these numbers in order from the least value to the greatest value, we get \( -5, -3, -1, 1, 3, 5 \)

Note that the difference between the successive predecessor and successor is 2

In general, if we denote the 1^{st} integer as \(n\), the consecutive even or consecutive odd integers will be \(n +2\), \(n +4\), \(n +6\), \(n +8\), ...

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**Solved Examples**

Example 1 |

Find the missing number in the series.

**\[3, 4, 5, 6, …, 8, 9, 10\]**

**Solution: **

The difference between any predecessor-successor pair in this series is 1

The predecessor of the missing number is 6

The successor of the missing number is 8

The missing number is **predecessor **\(+\)** difference** \(= 6 + 1 = 7\)

Alternatively, the missing number is **successor \(-\) difference** \(= 8 - 1= 7\)

\(\therefore\) The missing number in the series is 7 |

Example 2 |

Find the missing number in the series.

**\[4, 8, 12, ..., 20, 24, 28, 32\]**

**Solution: **

The difference between any predecessor-successor pair in this series is 4

The predecessor of the missing number is 12

The successor of the missing number is 20

The missing number is **predecessor **\(+\)** difference** \(= 12 + 4= 16\)

Alternatively, the missing number is **successor \(-\) difference** \(= 20 - 4= 16\)

\(\therefore\) The missing number in the series is 16 |

Example 3 |

What is the third number in the given series if they are all consecutive multiples of an odd integer?

**\[ 5, 15, ___ , 25, 30\]**

**Solution: **

The difference between any predecessor-successor pair in this series is 5

The predecessor of the missing number is 15

The successor of the missing number is 25

The missing number is predecessor\(+\) difference \(= 15 + 5= 20\)

\(\therefore\) The missing number is 20 |

Example 4 |

Find the missing numbers in the following series.

**\[75, …, 77, 78, …, 80\]**

**Solution: **

The difference between any predecessor-successor pair in this series is 1

The predecessor of the first missing number is 75

The successor of the first missing number is 77

The missing number is predecessor\(+\) difference \(= 75 + 1= 76\)

We see that numbers are in order: \(75, 76, 77, 78\) and the difference is \(1\)

Thus, the next number in the series will be \(79\)

\(\therefore\) Missing numbers in the series are 76 and 79 |

Example 5 |

The sum of three consecutive even numbers is 24.

What are the three numbers?

**Solution: **

Consecutive even numbers have a difference of 2 between them.

If the first number is \(n\), then the second number is \(n+2\) and the third number is \(n+4\).

Given that their sum is 24, hence, we have:

\(n + n+2 + n+4 = 24\)

\(\implies 3n + 6 = 24\)

\(3n = 24-6 = 18\)

\(\implies n = 6\)

Therefore, the numbers are

\(n =6\)

\(n+2 = 6+2 = 8 \)

\(n+4 = 6+4 = 10 \)

Let us add the three numbers and verify our solution.

\(10+8+6 = 24\)

\(\therefore\) The numbers are 6, 8, and 10 |

- Will the sum of two consecutive numbers be odd or even?
- Find the difference between consecutive square numbers. Do you see a pattern? What is their difference?
- Why is the product of 3 consecutive natural numbers always divisible by 6?

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**Practice Questions**

**Here are a few activities for you to practice. **

**Select/Type your answer and click the "Check Answer" button to see the result.**

- 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.

**Frequently Asked Questions(FAQs) **

## 1. What are 3 consecutive numbers?

Consecutive numbers are numbers that follow each other in order from the smallest number to the largest number.

Example 1, 2, 3 are the first three consecutive natural numbers.

## 2. How do you find consecutive numbers?

- Write the given numbers in order from the smallest to the largest.
- Find the difference between any predecessor-successor pair
- The missing number in the consecutive number list is
**predecessor \(+\) difference**

## 3. What are consecutive positive numbers?

Consecutive positive numbers are the set of positive numbers whose difference is 1.

\(1, 2, 3, 4, 5, 6...\) is the set of consecutive positive numbers.

## 4. Can consecutive numbers be decimals?

Consecutive numbers cannot be a decimal number because there exists several decimal number between every decimal number.

For example, if we say \( 3.1, 3.2 ,3.3... \) are consecutive numbers, several decimal numbers like \(3.11, 3.111, 3.1111 .... \) exists in between them.

## 5. What are 2 consecutive numbers?

Two numbers that follow each other in order are called 2 consecutive numbers.

Example: \(1,2\) are 2 consecutive natural numbers.

\(3,6\) are 2 consecutive multiples of 3.

\(10,20\) are 2 consecutive multiples of 10.