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Home >> Numbers >> Shortcut Method of Numbers >> Method : 2 >>

Shortcut Method of Numbers : Addition of Consecutive Odd Numbers

Method : 1 Method : 2 Method : 3

Before you understand this shortcut method, you are advice to read:

What are Odd Numbers ?
What are Natural Numbers ?
What are Square Numbers ?
Property 4 of Square Number

In some questions you are asked to add consecutive odd numbers which involves long calculation and time consuming too, So here is explained a short and easy method to solve such questions.
Before explaining the shortcuts, let's first understand, how you will solve such two examples as per normal process:

Example 1: Find Sum of first three odd numbers.
Solution: First three odd numbers are:
1, 3 & 5

Add these three odd number and we get:
1 + 3 + 5 = 9
Hence, Sum of first three odd numbers is equals to 9

Example 2: Find Sum of first five odd numbers.
Solution: First three odd numbers are:
1, 3, 5, 7 & 9

Add these three odd number and we get:
1 + 3 + 5 + 7 + 9 = 25
Hence, Sum of first five odd numbers is equals to 25

Shortcut Method

Now, above two examples are solved below in the shortcut ways:
And to understand this shortcut way, you are advice to read the property of square number which says that: If a Natural Number is a square number, then it has to be the sum of successive odd numbers starting from 1

So , with this property we get the following formula:
Sum of first n odd numbers = n2

Example 3: Find Sum of first three odd numbers.
Solution: Apply shortcut formula and we get:
Sum of first n odd numbers = n2

Since we have to find sum of three odd numbers, so here n = three or 3
Apply value of n in the above formula and we get:
Sum of first three odd numbers =32 = 3 X 3 = 9

Hence, Sum of first three odd numbers is equals to 9
(You can match the answers of example 1 and example 3)

Example 4: Find Sum of first five odd numbers.
Solution: Apply shortcut formula and we get:
Sum of first n odd numbers = n2

Since we have to find sum of first five odd numbers, so here n = five or 5
Apply value of n in the above formula and we get:
Sum of first five odd numbers =52 = 5 X 5 = 25

Hence, Sum of first five odd numbers is equals to 25
(You can match the answers of example 2 and example 4)

From comparing examples you can easily observe that you are saved from long calculation of adding each odd number one by one.

Shortcut method is very useful in case of big odd numbers (as in example 5)

Example 5: Find Sum of first twenty-five odd numbers
(If you solve this with normal process, it would very long and time consuming. So use shortcut method only)
Solution: Apply shortcut formula and we get:
Sum of first n odd numbers = n2
Since, we have to find sum of first twenty-five odd numbers, so here n = twenty-five or 25
Apply value of n in the above formula and we get:
Sum of first twenty-five odd numbers =252 = 25 X 25 = 625

Hence, Sum of first twenty-five odd numbers is equals to 625


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