Power's of 2 in pascal's triangle

Review

Pascal's Triangle

Pascal's Triangle is a triangular array of numbers where each number is the sum of the two numbers above it. It Looks something like this:

      1
     1 1
    1 2 1
   1 3 3 1
  1 4 6 4 1

And on and on...

Patterns and properties

Conbanations

Pascal's Triangle can also be written like this

And on and on... Remember that where .

Sum of rows

                1     =1
               1+1    =2
              1+2+1   =4
             1+3+3+1  =8
            1+4+6+4+1 =16

These are powers of two. Let's prove it true. (Note: There are dozens of more patterns but it would have nothing to do with powers of two).

Powers of two

Theorem

Theorem

It states that .

Why do we need it?

It is useful is many word problems (That means, yes, you can use it in real life) and it is just a cool thing to know. More at https://artofproblemsolving.com/videos/mathcounts/mc2010/419.

Proof

Short proof

Here is the short and sweet proof: If you look at the way we built the triangle you see that each number is row n-1 is added on twice in row n. This means that each row doubles. That means you get powers of two.

Which proof do you like better?

Which proof do you like better? The second one proves that you don't always have to brute the algebra right to the answer. When you're trying to prove something like this, step back, think about what your really trying to prove. Most of the time for me the answer just pops out.