Consecutive Numbers

Consecutive numbers are numbers that follow one another in order without skipping any values. If you start with any number and count upwards, each successive number will be one greater than the previous one. Similarly, if you count downwards, each successive number will be one less than the previous one.

Example set is given below:

Consecutive numbers are a sequence of numbers where each number follows the previous one in order, with a constant difference between them. Here are some examples to illustrate consecutive numbers in different contexts:

1. Consecutive Integers:

Example:

8, 9, 10, 11, 12

In this sequence, each number is one greater than the previous one. These are consecutive integers, and the common difference between them is 1.

2. Consecutive Even Numbers:

Example:

14, 16, 18, 20, 22

Here, each number is two greater than the previous one. These are consecutive even numbers, and the common difference is 2.

3. Consecutive Odd Numbers:

Example:

21, 23, 25, 27, 29

In this sequence, each number is two greater than the previous one. These are consecutive odd numbers, and the common difference is 2.

4. Consecutive Prime Numbers:

Example:

5, 7, 11, 13, 17 

Each number in this sequence is a consecutive prime number (a number greater than 1 with no positive divisors other than 1 and itself). The common difference between these numbers varies, as primes do not follow a regular pattern like even or odd numbers.

5. Consecutive Squares:

Example:

1, 4, 9, 16, 25

Here, each number is the square of the consecutive integers 1, 2, 3, 4, and 5. The common difference between consecutive squares is not constant.

6. Consecutive Cubes:

Example:

1, 8, 27, 64, 125

Here, each number is the cube of the consecutive integers 1, 2, 3, 4, and 5. The common difference between consecutive cubes is not constant

7. Consecutive Negative Integers:

Example:

-6, -5, -4, -3, -2

Even when working with negative numbers, consecutive numbers follow the same principle of each number being one greater than the previous one.

In general, when dealing with consecutive numbers, you can express the nth term of the sequence as an = a + (n – 1)d, where a is the first term, n is the position of the sequence and d is the common difference. Understanding consecutive numbers is fundamental in various mathematical concepts, including arithmetic progressions, algebra, and number theory.

Solved Examples

Question 1

Find the next three consecutive even numbers after 14.

Solution:

The consecutive even numbers after 14 are obtained by adding 2 successively: 14 + 2 = 16, 16 + 2 = 18, 18 + 2 = 20. So the numbers are 16, 18 and 20.

Question 2

The sum of three consecutive integers is 72. Find the integers.

Solution:

Let the three consecutive integers be n, n+1 and n+2.

Their sum = n + n + 1 + n + 2 = 72

=> 3n + 3 = 72

=> 3n = 69

=> n = 69/3 = 23

n + 1 = 23 + 1 = 24 and n + 2 = 23 + 2 = 25

So the three consecutive integers are 23, 24, and 25.

 Question 3

List five consecutive prime numbers greater than 20.

Solution:

A prime number is a number which is divisible by 1 and itself. So prime numbers greater than 20 are 23, 29, 31, 37 and 41.

Question 4

If 2x-1, 2x and 2x + 1 are consecutive odd integers, find their sum.

Solution:

Sum = (2x – 1) + 2x + (2x + 1) = 6x + 0 = 6x.

 Question 5

Find three consecutive integers such that the sum of the first two is 35 less than twice the third.

Solution:

Let the three consecutive integers be n, n+1 and n+2.

According to the question, n + (n + 1) = 2(n + 2) – 35

2n + 1 = 2n + 4 – 35

Subtracting 2n from both sides, we have

1 = -31

This equation has no solution, which means there are no consecutive integers that satisfy the given condition.

Consecutive Numbers FAQS

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