Consecutive Integers | Quantitative for GMAT PDF Download

Introduction

Consecutive integers are sequential numbers that follow one another in order without interruption. In mathematics, they form an unbroken sequence where each subsequent number is exactly one more than the preceding number. For instance, a set of natural numbers represents consecutive integers. Within this concept, the mean (average) and the median of a set of consecutive integers are the same. For any integer 'x', the consecutive integers are represented by 'x + 1' and 'x + 2'. This article delves into the detailed understanding of consecutive integers in mathematics, encompassing their meaning, properties, formulas, and illustrated examples.

Consecutive Meaning in Math

Consecutive meaning in Maths represents the unbroken sequence of numbers. It means that in a sequence, the numbers following continuously. To understand the consecutive meaning in Maths (i.e) consecutive numbers, first, we need to understand the concept of predecessors and successors. Predecessors mean that the number that is written immediately before the number. Whereas successors mean that the number that is written immediately after the number. Consider the sequence of numbers, 4, 5, 6, 7.  Here, the predecessor of 5 is 4 and the successor of 5 is 6. Thus, the consecutive numbers in Maths are the numbers that follow each other in order from the smallest number to the largest number.

Consecutive Integers in Maths

As discussed in the introduction, in Maths,  the numbers that follow each other in an order are called consecutive numbers or consecutive integers. These integers go from smallest to the highest, i.e. in ascending order.
Some of the examples are:

• -4, -3, -2, −1, 0, 1, 2, 3, 4
• 10, 11, 12, 13, 14, 15, 16
• 100, 101, 102, 103, 104, 105, 106
• 4, 5, 6, 7, 8
• -1, 0, 1, 2, 3, 4, 5, 6
• -20, -19, -18, -17

From the above examples, we can see, the integers follow each other in a sequence. The difference between preceding and succeeding integers is always equal to 1. 

• 4 - 3 = 1
• -2 - (-3) = 1
• 101 - 100 = 1

Odd Consecutive Integers

Consecutive odd integers are odd integers that follow each other and they differ by 2. If x is an odd integer, then x + 2, x + 4 and x + 6 are consecutive odd integers.
Examples:
5, 7, 9, 11,…
-7, -5, -3, -1, 1,…
-25, -23, -21,….

Even Consecutive Integers

Consecutive even integers are even integers that follow each other and they differ by 2. If x is an even integer, then x + 2, x + 4 and x + 6 are consecutive even integers. Consecutive even integers differ by two.
Examples:
4, 6, 8, 10, …
-6, -4, -2, 0, …
124, 126, 128, 130, ..

Consecutive Integers Formula

The given formulas are the algebraic representations of consecutive integers.
The formula to get a consecutive integer is n + 1,
For odd consecutive integers:
The general form of a consecutive odd integer is 2n+1,
For even consecutive integers:
The general form of a consecutive even integer is 2n,
Where
“n” can be any integer.

Product of Three Consecutive Integers

If there are three consecutive integers, say a, b and c, then their product is given by:
a × b × c = abc
Example:
1 × 2 × 3  = 6
7 × 8 × 9 = 504
9 × 10 × 11 = 990
From the above three examples, we can conclude an interesting fact that the product of any three consecutive integers, is always divisible by 6.

How to Find integers? 

Let us say, x, x + 1 and x + 2 are three consecutive integers, then the product of these three consecutive integers are given by:
x (x + 1) (x + 2) = x (x² + x + 2x + 2)
= x³ + 3x² +2x
By putting the above equation equal to the product of three consecutive integers and solving for x, we can determine the value of required integers.

Properties

The following are the properties of consecutive numbers:

• The gap between any two sequential odd or even integers is 2.
• In any set of n consecutive integers, there's exactly one number divisible by n. For instance, in a sequence of four integers, there will be a multiple of 4; in a series of 19 integers, there will be a multiple of 19. However, in a set of three consecutive integers like {-1, 0, +1}, the zero acts as the special case where it serves as the multiple of 3, given that multiplying any integer by zero yields zero.
• Depending on the starting point, a set of three consecutive integers may have two even numbers and one odd number or vice versa. In a sequence of four consecutive integers, there might be two even and two odd numbers. If a series contains an odd number of integers, there may be more even or odd numbers, while in a set with an even number of integers, the count of even and odd numbers will be equal.
• If m is an odd number, the total sum of m consecutive integers will be divisible by m. For instance, the sum of any five consecutive integers will be divisible by 5.

Consecutive Integers Problems

In Maths, there are many numerical and word problems that can be solved using this concept. Following are the example problems based on the concept of consecutive integers.

Example 1: Find three consecutive integers that add up to 51.
Ans: Suppose the three consecutive numbers are x, x + 1, x + 2
Given, sum of the numbers is equal to 51.
∴ x + x + 1 + x + 2 = 51
3x + 3 = 51
3x = 48
x = 16
Therefore,
x = 16,
x + 1 = 16 + 1 = 17,
x + 2 = 16 + 2 = 18
Thus, the numbers are 16, 17, 18.

Example 2: The sum of five consecutive integers is 100. find the third number.
Ans: Let the five consecutive integers be x, x + 1, x + 2, x + 3 and x + 4
As per the given questions,
X + x + 1 + x + 2 + x + 3 + x + 4 = 100
5x + 10 = 100
5x = 90
X = 18
Therefore, the third integer is x + 2 = 18 + 2 = 20