The sum of the n-th row of Pascal's Triangle is always 2 to the power of n.

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.

The binomial coefficients

The translation of the word binomial means "two names". In algebra, a polynomial is an expression with several terms, a term being a set of numbers and variables multiplied together. The number in the term is called the coefficient.  Here are some examples

7xyz² is a single term with three variables. 7 is the coefficient, Since the z is raised to the second power, if we wrote this out in letters it would be 7xyzz, and since we have to used four letters to write this out completely, we say this is a fourth degree term.

7xy - 2z² is two terms, so it is called a binomial. The coefficient of the first term is 7 and the coefficient of the second term is -2.

What happens when we raise a binomial to different powers? Here are the first few examples.

(x + y)º = 1
(x + y)¹ = x + y
(x + y)² = x² + 2xy + y²
(x + y)³ = x³ + 3x²y + 3xy² + y³

If we strip away the variables and just look at the coefficients, we get

1
1 1
1 2 1
1 3 3 1

If you have been reading the blog this week, you know those are the first few rows of Pascal's triangle.

Here is the standard summation form of this fact, usually called The Binomial Theorem.

  
Notice that the powers for x and y always add up to n when we raise (x + y) to the n-th power.

You might also notice that the sums of the rows of Pascal's Triangle has an easily recognizable pattern.

1 = 1
2 = 1+1
4 = 1+2+1
8 = 1+3+3+1

In these first few cases, it looks like the sum of the n-th row is 2 raised to the power of n. There are many ways to prove this pattern continues forever. 

Tomorrow, we will use The Binomial Theorem to prove this pattern.