Difference between revisions of "Power's of 2 in pascal's triangle"
| Line 20: | Line 20: | ||
<math>\binom{0}{0}</math> | <math>\binom{0}{0}</math> | ||
<math>\binom{1}{0}</math> <math>\binom{1}{1}</math> | <math>\binom{1}{0}</math> <math>\binom{1}{1}</math> | ||
| − | <math>\binom{2}{0}</math> <math>\binom{2}{1}</math> <math>\binom{2}{ | + | <math>\binom{2}{0}</math> <math>\binom{2}{1}</math> <math>\binom{2}{2}</math> |
And on and on... | And on and on... | ||
| Line 52: | Line 52: | ||
=== Short 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? === | ||
| + | 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. | ||
Revision as of 15:15, 16 June 2019
Contents
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
Long 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.
