Question 6
Let $[x]$ denote the integer part of $x$, i.e., the greatest integer not exceeding $x$. If $n$ is a positive integer, express as a simple function of $n$ the sum
$$\left\lfloor\frac{n+1}{2}\right\rfloor+\left\lfloor\frac{n+2}{4}\right\rfloor+\cdots+\left\lfloor\frac{n+2^i}{2^{i+1}}\right\rfloor+\cdots.$$
$$\left\lfloor\frac{n+1}{2}\right\rfloor+\left\lfloor\frac{n+2}{4}\right\rfloor+\cdots+\left\lfloor\frac{n+2^i}{2^{i+1}}\right\rfloor+\cdots.$$