Yesterday, we discussed the Binomial Theorem, the standard way to expand a binomial, here expressed as (

*x*+

*y*), raised to any positive integer power

*n*.

We also have a simple pattern easily visible in the first few rows of Pascal's Triangle.

1 = 1

2 = 1+1

4 = 1+2+1

8 = 1+3+3+1

16= 1+4+6+4+1

...

From what we can see, the sum of the

*n*-th row is 2 raised to the power of

*n*. In math, we can't just say we see a pattern for a few rows and assume it is always true, we need to

*prove*it is true. Here is the proof.

Step 1 is no more difficult than 2 = 1+1. Pretty easy.

Step 2 is to take the Binomial Theorem and replace the general binomial (

*x*+

*y*) with the specific binomial (1+1). So we get all the entries of row

*n*multiplied by powers of 1 then added up.

Any integer power of 1 is 1, so we can erase those powers because they are superfluous.

Now we just have the sum across the row. Since the equations are of the form

*a*=

*b*, then

*b*=

*c*, then

*c*=

*d*, we can also say

*a*=

*d*.

This is not the only way to prove this. Another proof of this basic property of Pascal's Triangle tomorrow.

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