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 2 of 5: Scale the identity
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[n2k]=[n2k+1]+[n+2k2k+1]\left[\frac n{2^k}\right]=\left[\frac n{2^{k+1}}\right]+\left[\frac{n+2^k}{2^{k+1}}\right]
Detailed analysis

Apply the identity to x=n/2kx=n/2^k. This gives the displayed relation for every integer k≥0k\ge0.