Formula for the n-th row of Pascal's Triangle

[stgermaine]

Homework Statement

Find a formula for the sum of the elements of the nth row of Pascal's Triangle

Homework Equations

C(n,r) = [C(n-1,r-1) + C(n-1,r)]
C(n,0) = C(n,n) = 1

The Attempt at a Solution

I started with the summation of the elements in the rows n
[itex]\sum^{n}_{r=0} C(n,r)[/itex]
= [itex]C(n,0) + \sum^{n-1}_{r=1} C(n,r) + C(n,n)[/itex]
= [itex]C(n,0) + \sum^{n-1}_{r=1} [C(n-1,r-1) + C(n-1,r)] + C(n,n)[/itex]
= [itex]C(n,0) + \sum^{n-1}_{r=1} C(n-1,r-1) + \sum^{n-1}_{r=1} C(n-1,r) + C(n,n)[/itex]
= [itex]C(n,0) + \sum^{n-2}_{r=0} C(n-1,r) + \sum^{n-1}_{r=1} C(n-1,r) + C(n,n)[/itex]
= [itex]C(n,0) + [C(n-1,0) + \sum^{n-2}_{r=1} C(n-1,r)] + [\sum^{n-2}_{r=1} C(n-1,r) + C(n-1,n-1)] + C(n,n)[/itex]

= [itex]2 \cdot C(n-1,0) + \sum^{n-2}_{r=1} C(n-1,r) + \sum^{n-2}_{r=1} C(n-1,r) + 2 \cdot C(n-1,n-1)[/itex] since C(n,0) = 1 = C(n-1,0) and C(n,n) = 1 = C(n-1,n-1)
=[itex]2 \cdot \sum^{n-1}_{r=0} C(n-1,r)[/itex]

This gives the intuitively correct response that the sum of the nth row is twice that of the (n-1)th row. How can I go from here to sum = 2^n?

Thank you!

 

[jbunniii]

Do you know the binomial theorem? What is a formula for [itex](x+y)^n[/itex] in terms of [itex]C(n,r)[/itex]?

 

[jbunniii]

This gives the intuitively correct response that the sum of the nth row is twice that of the (n-1)th row. How can I go from here to sum = 2^n?

This should be a simple induction. What is the sum of the 1st row?