Defining Consecutive Integers – Natural, Even and Odd
In lower grades, students are introduced to the concepts of positive natural numbers, negative integers and whole numbers starting with zero. There was also an introduction to even and odd numbers. The unique aspect of these numbers is that they all follow a sequence.
Natural numbers: 1, 2, 3, 4, 5… 567, 568, 569… are examples of consecutive natural numbers.
Whole numbers: 0, 1, 2, 3, 4… 458, 459, 460… are examples of consecutive whole numbers.
Even numbers: 2, 4, 6, 8, 10… 678, 680, 682… are examples of consecutive even numbers.
Odd numbers: 1, 3, 5, 7, 9, 11… 345, 347, 349… are examples of consecutive odd numbers.
Negative integers: -1, -2, -3, -4… -245, -246, -247… are examples of consecutive negative integers.
What Are the Formulas Linked to Consecutive Numbers?
There are a few formulas that are associated with consecutive numbers. We can start with the consecutive natural numbers.
If ‘n’ is a positive natural number, then the next few numbers in this sequence are n, n + 1, n + 2, n + 3, n + 4…
With reference to even numbers, the formula associated with such numbers is: 2n, 2n + 2, 2n + 4, 2n + 6, 2n + 8…
Moving on to odd numbers, the formula associated with odd numbers is: 2n + 1, 2n + 3, 2n + 5, 2n + 7, 2n + 9…
Rapid Recall
Solved Examples
Example 1:
Given that the sum of three consecutive natural numbers is 201, what will be the product of these numbers?
Solution:
Given in the problem:
Sum of three consecutive natural numbers = 201
Let x be the first number, x + 1 is the second and x+2 the third number.
So, \(x + x + 1 + x + 2 = 201\) [Write the equation]
\(\Rightarrow 3x + 3 = 201\) [Simplify equation]
\(\Rightarrow 3x + 3 – 3 = 201 – 3\) [Subtract 3 from both sides]
\(\Rightarrow 3x = 198 \) [Simplify]
\(\Rightarrow x = 66 \) [Divide each side by 3]
So, the first number in the sequence is 66, the next number is 67 and the last number is 68.
Product of these numbers \(= 66 \times 67 \times 68\) [Write the expression]
\(\Rightarrow 300696\)
Hence the product of the three consecutive numbers, 66, 67 and 68 is 300696.
Example 2:
Given that the average of four consecutive even numbers is 9. Find the numbers and the product of these four even numbers.
Solution:
Let the even numbers be, 2n, 2n + 2, 2n + 4, 2n + 6
Given in the problem: Average of numbers = 9
\(\frac{2n + 2n + 2 + 2n + 4 + 2n + 6}{4} = 9\) [Write the equation]
\(\Rightarrow \frac{8n + 12}{4} = 9\) [Simplify the equation]
\(\Rightarrow 8n = 36 – 12\) [Simplify]
\(\Rightarrow 8n = 24\) [Simplify]
\(\Rightarrow n = 3\)
So, the first even number is 6, followed by 8, 10 and 12.
Product of these numbers \(= 6 \times 8 \times 10 \times 12\)
\(\Rightarrow 5760\)
Hence, the product of the four consecutive even numbers is 5760.
Example 3:
Jean and her two friends have 267 marbles that they need to divide among themselves such that they receive the marbles as a set of three consecutive odd numbers. How many marbles does each friend receive?
Also find the difference between the highest and lowest number of marbles.
Solution:
Given in the problem: 267 marbles need to be distributed among three friends in the sequence of three odd numbers.
Let the odd numbers be: 2n + 1, 2n + 3 and 2n + 5
\(2n + 1 + 2n + 3 + 2n + 5 = 267\) [Write the equation]
\(\Rightarrow 6n = 258\) [Simplify]
\(\Rightarrow n = 43\) [Isolate the value of n]
So the odd numbers are, 87, 89, 91
Jean and her two friends received 87, 89 and 91 marbles each.
Difference between the highest and lowest number of marbles = 91 – 87 = 4
Hence, the difference between the highest and the lowest number of marbles is 4 marbles.
When referring to positive integers, they are observed on the right of zero on a number line. Negative integers are placed to the left of the zero on a number line and are lesser than zero.
Consecutive integers are numbers that occur in a specific sequence. Consecutive integers may be natural numbers, odd numbers and even numbers.
Prime numbers do not occur in any particular sequence as compared to the sequence seen in natural numbers, even numbers and odd numbers.