Let $N = \sum_{k = 1}^{1000}k(\lceil \log_{\sqrt {2}}k\rceil - \lfloor \log_{\sqrt {2}}k \rfloor).$ Find $N$.

Question

Let$$N = \sum_{k = 1}^{1000}k(\lceil \log_{\sqrt {2}}k\rceil - \lfloor \log_{\sqrt {2}}k \rfloor). $$ Find $N$.

I'm not sure how to start this problem. Could someone help me out? Thanks!

Answer 1

HINTS:

Note that we have

$$\lceil x\rceil-\lfloor x\rfloor =\begin{cases}1&,x\notin \mathbb{Z}\\\\0&,\text{otherwise}\end{cases}$$

In addition, $\log_{\sqrt 2}(k)\in \mathbb{Z}$ only when $k$ takes on values $2^n$, for $n=0, 1, \dots, 9$.

And now you can finish.