Last modified on August 3rd, 2023

chapter outline

 

Consecutive Integers

Consecutive integers follow one after the other in an ordered sequence. Thus they form an unbroken chain of numbers in ascending order, where the next number is one more compared to the previous one.

For example, a set of natural numbers such as 1, 2, 3, 4, and 5 are consecutive integers. Whenever we count any items, we use consecutive integers.

Thus, if n is an integer, then the consecutive integers are given as {n, n + 1, n + 2, n + 3,….} 

Consecutive integers can be of even odd, even, positive, or negative numbers.

Consecutive Odd Integers

They are odd integers that follow each other in a sequence, so the difference between them is 2. If n is an odd integer, then n + 2, n + 4, and n + 6 are the consecutive odd integers.

A set of consecutive odd integers is written as {n, n + 2, n + 4, n + 6,….}

Examples

  • 3, 5, 7, 9,…
  • -5, -7, -3, -1,…

Consecutive Even Integers

They are even integers that follow each other in a sequence such that the difference between them is 2. If n is an even integer, then n + 2, n + 4, and n + 6 are the consecutive even integers.

A set of consecutive even integers is written as {n, n + 2, n + 4, n + 6, …..}

Examples

  • 2, 4, 6, 8,…
  • -6, -4, -2, 0,…
  • 100, 102, 103, 104,…

Formulas

The general formulas, where n is an integer, are:

Consecutive Integers

Properties

  1. The difference between the consecutive integers in a sequence is always the same. For example, in the sequence -4, -3, -2, -1, 0, 1, 2, 3, 4, the difference between each successive number is 1
  2. The difference between the consecutive odd integers in a sequence is always 2. For example, in the sequence -7, -5, -3, -1, 1, 3, 5, 7, the difference between each successive number is 2
  3. The difference between the consecutive even integers in a sequence is always 2. For example, in the sequence -8, -6, -4, -2, 0, 2, 4, 6, 8, the difference between each successive number is 2
  4. If n is an odd integer, then the total sum of n consecutive integers will be divisible by n. For example, for any 7 integers in a row, the sum is divisible by 7

Consecutive Positive Integers

Consecutive positive integers include a sequence of natural numbers having a fixed difference between them. For example, {1, 2, 3, 4, 5,…} is a set of consecutive positive integers with a difference of 1. Consecutive positive integers can be consecutive odd positive integers or even positive integers.

3 Consecutive Integers

3 consecutive integers are a sequence of 3 integers, so the difference between them is fixed. They are commonly used to solve problems in mathematics.

For any 3 consecutive integers, say n, n + 1, and n + 2:

Sum

n + (n + 1) + (n + 2)

Example

3 + (3 + 1) + (3 + 2)

Product

n × (n + 1) × (n + 2) = n(n2 + n + 2n + 2)

Example

3 × 4 × 5

Solved Problems

Assume that the sum of three consecutive even integers is 108. What is the largest number?

Solution:

Given, the sum of 3 consecutive even integers is 108
Let the 3 consecutive even numbers be 2n, 2n + 2, and 2n + 4
By adding, we get,
2n + (2n + 2) + (2n + 4) = 108
=> 2n + 2n + 2n + 2 + 4 = 108
=> 6n + 6 = 108
=> 6(n + 1) = 108
=> n + 1 = 18
=> n = 17
Substituting the value of x, we get,
2n = 2(17) = 34
2n + 2 = 34 + 2 = 36
2n + 4 = 34 + 4 = 38
Therefore, the largest number is 38

Find 3 consecutive integers that add up to 45

Solution:

Let the 3 consecutive integers be n, n+1, n+2
Given, the sum of the integers is equal to 45
∴ n + (n + 1) + (n + 2) = 45
=> 3n + 3 = 45
=> 3n = 42
=> n = 14
Therefore,
n = 14,
n + 1 = 14 + 1 = 15,
n + 2 = 14 + 2 = 16
Thus, the integers are 14, 15, 16

The sum of two consecutive integers is 55. Find the integers?

Solution:

Let the 2 consecutive integers be n, n + 1
Given, the sum of the integers is equal to 55
∴ n + (n + 1) = 55
=> 2n + 1 = 55
=> 2n = 54
=> n = 27
n + 1 = 27 + 1 = 28
Thus, the integers are 27, 28

The sum of 5 consecutive integers is 265. Find the integers.

Solution:

Let the numbers be n, n + 2, n + 4, n + 6 and n + 8
Given,
The sum of the integers is equal to 265
∴ n + (n + 1) +( n + 2) + (n + 3) + (n + 4) = 265
=> 5n + 10 = 265
=> 5n = 255
=> n = 51
n + 1 = 51 + 1 = 52
n + 2 = 51 + 2 = 53
n + 3 = 51 + 3 = 54
n + 4 = 51 + 4 = 55
 So, the integers are 51, 52, 53, 54

The product of two consecutive even integers is 72. What are the numbers?

Solution:

Let the numbers be n, n + 1
Given product of two consecutive integers is 72
∴ n(n + 1) = 72
=> n2 + n = 72
=> n²+ n -72 = 0
=> (n + 9)(n – 8)
n = {8, -9}
Since, n = -9 is not possible, n = 8 (first number)
The next number = 8 + 1 = 9
Thus the two consecutive even integers are 8 and 9

Last modified on August 3rd, 2023

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