# Expressing Square Numbers as Sum of consecutive Natural Numbers

You can express any odd perfect square in terms of the sum of two consecutive natural numbers. For example, 3 squared or 9 can be expressed as (4 + 5). 5 squared or 25 can be expressed as (12 + 13) and so on.

What this means, is that we can express the square of any odd number as the sum of two consecutive natural numbers.

We cannot do this for the square of even numbers because the square of an even numbers is always an even number. An example would be 16. 16 is a perfect square and it can be expressed as (8 + 8). 8 and 8 are NOT two consecutive natural numbers but (12 + 13) are two consecutive natural numbers added together to get to 25 which is a perfect square.

One important thing to notice here is that the inverse of the rule above is not always true, so it’s not true at all. Meaning that, (12 + 13) would be 25 which is a perfect square but let’s pick to other consecutive natural number like (13 + 14) which is 27. 27 is not a perfect square. So while any odd square number can be expressed as the sum of two consecutive natural numbers, not any two consecutive natural numbers added together would result in an odd perfect square.

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