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 1 of 5: Use the basic floor identity
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[x]=[x2]+[x+12][x]=\left[\frac x2\right]+\left[\frac{x+1}{2}\right]
Detailed analysis

Write x=2q+rx=2q+r with qq an integer and 0≤r<20\le r<2. Checking the two ranges 0≤r<10\le r<1 and 1≤r<21\le r<2 proves the identity.