Comment by tomlor

I came at a solution doing the following:

Well, I know this:

   1 = 1/2 + 1/2
   

We're trying to get to 5 terms. So let's split 1/2 into two terms.

   1 = 1/2 + 1/4 + 1/4
   

Hmm, let's do that again - this time splitting 1/4.

   1 = 1/2 + 1/4 + 1/8 + 1/8
   

Ok, I'm pretty close. Can I split 1/8 into 1/9 and some mystery term?

   1 = 1/2 + 1/4 + 1/8 + 1/9 + 1/N
   

Well, that's not hard to solve. N = 72.

1 = 1/2 + 1/4 + 1/8 + 1/9 + 1/72