I can't make out the image in your post, but based on your description of the problem, it should help to know that the sum of the elements in the nth row of Pascal's triangle is . The standard way to prove this, in case you don't already know, is to start with the binomial theorem,
and then set x=1, with the result
That doesn't solve your problem completely, but it should be a good first step. Maybe you can do the rest.