The question I'm looking at, is to show that every positive integer $n$ can be written as a sum of distinct powers of two.
I can see that you can form any number based on the highest $2^t$ that is less than the number, plus some combination of $2^j<n$'s. And that you can make the number odd, by adding $2^0$ at the end.
I just don't know how to create the formula for the proof. I'm trying to figure out my base case, and then my inductive formula to figure out $k+1$, and I've got nothing.