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Let 

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

 Mar 15, 2018
 #1
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∑[(k*(ln(k)/0.34657359)) - (ln(k)/0.34657359)], for k=1 to 1000

N≈9.23734613867.....×10^6

Note: I used natural log(ln) instead of common log(base 10), but it should not alter the result.

Also I converted log(sqrt(2)) to a constant of 0.34657359. Check my result by summing up 3-4 terms using the notation above. I did so and it seems OK.

 Mar 15, 2018
edited by Guest  Mar 15, 2018
 #2
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I may have misread your question!! I multiplied the constant k, by the first term only. Looking at it more closely, it appears that it should be multiplied by the whole expression as follows:

∑k*[(ln(k)/0.34657359) - (ln(k)/0.34657359)], for k=1 to 1000

 

If this is accurate, then N = 0 !!!.

 Mar 15, 2018

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