Expressing Square Numbers as Sum of Odd Natural Numbers

There is a relationship between the number of odd natural numbers starting from 1 added together and square numbers. That means, 1, the same things as 1 squared is the same thing as the first odd natural number. And you know that 1 is a perfect square. 4, which is the same thing as 2 squared, can be written as (1 + 3) which is the first two natural odd numbers starting from 1 added together. 9, which is the same thing as 3 squared, can be written as (1 + 3 +5), which is the same thing as the first three natural odd numbers starting from 1 added together and so on and so forth.

What this means is that if you add the first n natural odd numbers starting from 1 together, the result can be expressed as n squared.

You can use this observation in different ways in different situations to check whether a number a perfect square or not. For example, if you were to test whether 121 is a perfect square, you’d check whether 121 can be expressed as a sum of some odd natural numbers starting from 1. If you tried this, you’d find out that the number of those odd natural numbers, in this case, would be 11, namely 1 through 21 or 1, 3, 5, 7, 9, … , 21. If you add all those numbers together, you’d get 121. So you can conclude that 121 is perfect square. Moreover, sqrt(121) = 11.

Author: Gahara Raschedian

Math and Science Teacher

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