Pascal’s Triangle

How to find the formulae for a+b to the power 4 or 5 or more

Pascal’s Triangle is a very simple construction out of natural numbers. We simply add up two numbers in the previous row and write down below the two numbers. 

The first and the last numbers are always 1. 

There are many amazing properties of these numbers

121 is 11 squared. 1331 is 11 cubed & 14641 is 11^4

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Pascal’s Triangle helps us arrive at the expansion of (a+b) to any power.

We know the formulae for (a+b)^2 and (a+b)^3

Now as shown in the diagram, we can find (a+b)^4 or (a+b)^5  or any other power

Pascal’s triangle numbers in each row gives us the coefficients of the terms of the expansion.

And the terms themselves can be simply derived. For example in the expansion of (x+y)^4, the coefficients are 14641 while in the terms, the power of x comes down from 4 to 3 to 2 to 1 to 0 while the powers of y goes up from 0 to 1 to 2 to 3 to 4. 

Similarly for (x+y)^5. The coefficients are 1 5 10 10 5 1 and the power of x comes down from 5 to 0 & powers of y goes up from 0 to 5


Vignettes from Nature