MathLabs

Problem 6

Let [x][x] denote the greatest integer not exceeding xx. If nn is a positive integer, express [n+12]+[n+24]+[n+48]+⋯[\frac{n+1}{2}]+[\frac{n+2}{4}]+[\frac{n+4}{8}]+\cdots as a function of nn.
Step 3 of 5: Recognize the telescoping terms
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[n+2k2k+1]=[n2k]−[n2k+1]\left[\frac{n+2^k}{2^{k+1}}\right]=\left[\frac n{2^k}\right]-\left[\frac n{2^{k+1}}\right]
Detailed analysis

Rearrange the scaled identity. The summand in the problem is exactly the difference of two successive terms in the floor sequence.