Addition of Consecutive Odd numbers is equal to Square Numbers

by Francis 2012-12-01 17:23:02

Addition of Consecutive Odd numbers is equal to Square Numbers
square-table.png

In this Square Table To Explain following Properties,
1. Consecutive Odd number Addition is equal to consecutive square value.
2. A square number previous odd number is to multiply that number by 2 and separate by 1
3. A square number next odd number is to multiply that number by 2 and add by 1

Example
If you find 82,
Previous Odd Number is, 2*8 - 1 = 15
So that, 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64

Square Formula:
square.png
Proof:
L.H.S = [((a2 + b2 + 2ab) + (a2 + b2 - 2ab) ) / 2] - b2
= [(2a2 + 2b2) / 2] - b2
= [2(a2+b2) / 2] - b2
= (a2+b2) - b2
= a2
= R.H.S
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