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Prove that for n=1, 2, 3... [(n+1)/2]+[...

Prove that for `n=1, 2, 3...` `[(n+1)/2]+[(n+2)/4]+[(n+4)/8]+[(n+8)/16]+...=n` where `[x]` represents Greatest Integer Function

Text Solution

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`[x]=[x/2]+[(x+1)/2]`
`[n]=[n/2]+[(n+1)/2]`
`[n/2]=[n/4]+[(n+2)/4]`
`[n/4]=[n/8]+[(n+4)/8]`
`[(n+1)/2]+[(n+2)/4]+[(n+4)/8]+...=[n]`.
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