100 followers problem-2

Algebra Level 3

log1002+log10024+log10028+log100216+\large\log_{100}{\sqrt{2}}+\log_{100}{\sqrt[4]{2}}+\log_{100}{\sqrt[8]{2}}+\log_{100}{\sqrt[16]{2}}+\ldots Let AA denote the value of series above. Find the value of 100A\lfloor100A\rfloor.

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