Consecutive Integers Formula

Consecutive Integers Formula

The Consecutive Integers Formula is used to solve a variety of mathematical issues. N, n+1,… n+k is the Consecutive Integers Formula. Integers that follow one another in a predictable counting pattern are referred to as consecutive integers. There are no numbers missed while listing consecutive integers in a sequence, so the difference between them is always fixed. The following integers in increasing order are referred to as consecutive integers. The Consecutive Integers Formula can be used to determine if a group of integers is consecutive or not, or to find such integers for any given number.

Consecutive integers are represented algebraically by the Consecutive Integers Formula.

N + 1 is the Consecutive Integers Formula to obtain a consecutive integer.

For a series of odd integers:

A string of successive odd numbers has the generic form 2n+1,

For a series of even integers:

A string of successive even integers has the general form 2n.

Where

Any integer can be “n.”

For a better understanding of the Consecutive Integers Formula, students can sign up on the Extramarks website.

What are Consecutive Integers?

A series of Consecutive Integers Formula is represented by the formula n, n+1,…n+k. The numbers that come after one another are known as consecutive integers. They proceed sequentially or alphabetically. For instance, a group of natural numbers is a sequence of integers. In Mathematics, the term “consecutive” refers to an uninterrupted succession or a continuous flow of numbers, so that successive integers follow a sequence in which each succeeding number is one more than the one before it. The mean and median in an array of successive integers (or in numbers) are both identical. In the event that x is an integer, then x + 1 and x + 2 are also integers. Numerous mathematical issues can be solved using the Consecutive Integers Formula. Learning resources on the Consecutive Integers Formula can be found on the website and mobile application of Extramarks.

Consecutive Even Integers

Even integers that are consecutive and differ by 2 are known as even integers. If x is an even number, then x + 2, x + 4, and x + 6 are also even numbers that follow each other. Even integers that follow one another differ by two. Students can also sign up on the Extramarks website for a better understanding of the concepts related to the Consecutive Integers Formula.

Examples:

4, 6, 8, 10, …

-6, -4, -2, 0, …

Consecutive Odd Integers

Consecutive odd numbers are odd integers that are separated by 2 and come after one another. If the number x is odd, then the numbers x + 2, x + 4, and x + 6 are also odd.

Examples:

5, 7, 9, 11,…

-7, -5, -3, -1, 1,…

Consecutive Integers Formula

According to the definition of the Consecutive Integers Formula given in the previous sections, we get to the conclusion that consecutive integers have the following form: x, x + 1, x + 2, x + 3,…, where x is an integer and x + 1, x + 2,.. are subsequent Consecutive Integers Formula in sequence. The first integer in a problem involving sequential integers is assumed to be x, and the succeeding integers can be derived by adding 1 to the prior integer. N, n+1,… n+k is the Consecutive Integers Formula. Students can refer to the Consecutive Integers Formula available on the Extramarks website. 

The difference between two successive even (or odd) integers is 2, as is known. Since x is an even/odd integer and x + 2, x + 4,.. are successive even/odd consecutive integers in succession, any two consecutive even integers (or) consecutive odd integers are of the form: x, x + 2, x + 4.

Properties of Consecutive Integers

Integers that come after one another in ascending sequence are said to be consecutive. The characteristics of successive integers should be noted.

Each succeeding integer in a sequence has the same difference. Look at the list below as an illustration: -2, -1, 0, 1, 2. The difference between each succeeding number is 1, as can be seen. As a result, if we assume that the first integer is “x,” the series can be expressed as “x,” “x +1,” “x + 2,” “x + 3,” and so on. 

Each subsequent odd integer has a difference of two. A sequence of consecutive odd numbers, such as -3, -1, 1, 3, 5, 7, and so on, reveals that there is a 2-unit difference between each next number. (7 – 5 = 2)

There is always a 2 space between each succeeding even integer. A sequence of consecutive even integers, such as -4, -2, 0, 2, 6, 8, 10, 12, and so on, reveals that there is a 2-unit difference between each next number. (10 – 8 = 2)

Always divide the total of ‘n’ consecutive odd integers by ‘n’. For instance, any two successive odd numbers added together will always have a sum that can be divided by two. Similar to this, any 15 consecutive odd numbers added together are always divisible by 15.

Consecutive Positive Integers

A series of natural numbers with a fixed difference is referred to as the Consecutive Integers Formula. For instance, the positive integers 1, 2, 3, 4, 5,… have a fixed difference of 1 between them. Numerous Consecutive Integers Formula sequences are possible, including consecutive even positive integers and consecutive odd positive integers. 

Find two successive positive integers whose total squares are 365, for instance.

The difference between two successive positive numbers is 1, so if one integer is assumed to be x, the other must be x + 1.

The formula is x2 + (x + 1)2 = 365.

[Using the algebraic identity (a + b)2 = a2 + 2ab + b2] x2 + x2 + 1 + 2x = 365

⇒ 2×2 + 2x + 1 = 365

⇒ 2×2 + 2x + 1 – 365 = 0

⇒ 2×2 + 2x – 364 = 0

⇒ x2 + x – 182 = 0

⇒ x2 + x – 182 = 0

⇒ x2 + 14x – 13x – 182 = 0

⇒ x(x + 14) – 13 (x + 14) = 0

⇒ (x – 13) (x + 14) = 0

⇒ x = 13, or x = -14

Since a positive integer is required, x = -14 is disallowed. So, x = 13.

Then, x + 1 = 14

Answer: 13 and 14 are the required consecutive positive integers.

Three Consecutive Integers

A series of three integers where their difference is fixed is said to be the Consecutive Integers Formula. To solve problems based on consecutive integers, typically three consecutive integers are found given under a specific condition.

There is a case study to better comprehend this

Find three consecutive integers whose sum is 51, for instance.

Solution: Assume that x is the first integer and that the subsequent two are x + 1 and x + 2.

The equation is x + (x + 1) + (x + 2) = 51.

⇒ x + x + 1 + x + 2 = 51

⇒ 3x + 3 = 51

⇒ 3(x + 1) = 3 × 17

⇒ x + 1 = 17

⇒ x = 17 – 1

⇒ x = 16

The other two numbers are 16 + 1 and 16 + 2, respectively.

Answer: 16, 17, and 18 are the necessary integers. 

Consecutive Integers Examples

Find the missing numbers in the following sequence using the property of consecutive integers: 7, 14, 21, _, 35, _, and 49.

Solution:

A difference of 7 exists between each number in the given series of 7, 14, 21, _, 35, _, and 49. Therefore, the missing numbers can be found by applying this feature of consecutive integers. Let’s add 7 to 21 since it comes before the first missing number to produce the following integer. 21 + 7 = 28. Let’s now verify that by calculating the difference between the number 28 and its successor. 35 – 28 = 7.

Let’s multiply 35 by 7 to obtain the second missing integer, which is the predecessor to the first missing number. 35 + 7 = 42. Let’s now verify that by determining the difference between 42 and its replacement. 49 – 42 = 7.

The two integers 28 and 42 are needed.

The Extramarks’ website provides study material to students on the Consecutive Integers Formula in detail. 

Consecutive Integers Questions

1. Find the missing number in the following series using the property of consecutive integers: 3, 6, _, 12, 15, and 18.

Options- 9, 10, 11 

The answer is 9. 

1. Write the first three consecutive odd integers using the property of consecutive integers.

Options-

1, 2, 3

3, 5, 7

5, 7, 9

Solution- 3, 5, 7