Question

對(duì)于任意實(shí)數(shù)x,符號(hào)［x］表示求不超過(guò)x的最大整數(shù).例如,［-1.3］=-2,［π］=3,［0］=0,那么［log₂1］+［log₂2］+［log₂3］+…+［log₂512］等于

A.3 595                B.3 586                C.1 547                D.1 555

Answer 1

A  由［x］的含義知:［log₂1］+［log₂2］+［log₂3］+…+［log₂512］

=0+2×1+2²×2+2³×3+…+2⁸×8+9.記S=1×2+2×2²+3×3³+…+8×2⁸,

則2S=1×2²+2×2³+…+7×2⁸+8×2⁹,∴-S=2+2²+2³+…+2⁸-8×2⁹=-8×2⁹=-3 586.

∴S=3 586.故原式=3 586+9=3 595.