All of us know what (a+b)² formula is, But sometimes even these most basic things have some neat ideas hidden inside them. This post explains the proof of (a+b)² from a geometric perspective(age old) and tries to appreciate its hidden beauty. To start with, lets take a square and divide its length/breadth into a units and call the remaining length b units as shown in the below figure. Now it is evident from the figure
Total Area = Area of blue square + top green rectangle + right green rectangle + purple square
This proof may be easily extended to prove (a+b)³ by imagining volumes of cubes instead of areas. But clearly, for any higher dimension > 3 we might not be able to explain it geometrically and we may have to fall back to abstract ideas of algebra.