Find a closed form

Question

How do I prove (with strong induction) that every positive integer $n$ has a representation in the form $$n = c_r2^r + c_{r−1}2^{r−1} + \cdots + c_2 2^2 + c_1 2 + c_0$$ where $r$ is a nonnegative integer, $c_r = 1$, and $c_j = 1$ or $0$ for all $j = 0, 1, 2, \ldots , r − 1$

Thanks

Answer 1

If $n$ is even, consider $n/2$. If $n$ is odd, consider $(n-1)/2$. By “considering” a number, I mean apply the induction hypothesis to it.