100 followers problem-2

Algebra Level 3

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

The answer is 15.

1 solution

Chew-Seong Cheong
Jun 1, 2015

$\begin{aligned} A & = \log_{100} {\sqrt{2}} + \log_{100} {\sqrt[4]{2}} + \log_{100} {\sqrt[8]{2}} + \log_{100} {\sqrt[16]{2}} + ...\\ & = \frac{1}{2} \log_{100} {2} + \frac{1}{4} \log_{100} {2} + \frac{1}{8} \log_{100} {2} + \frac{1}{16} \log_{100} {2} + ... \\ & = \frac{1}{2}\left(\frac{1}{1-\frac{1}{2}} \right)\log_{100} {2} = \log_{100} {2} \end{aligned}$

$\begin{aligned} \Rightarrow \lfloor 100A \rfloor & = \left \lfloor 100 \log_{100} {2} \right \rfloor = \left \lfloor 100 \times \frac {\log_{10} {2}} {\log_{10} {100}} \right \rfloor \\ & = \left \lfloor 100 \times \frac {0.3010} {2} \right \rfloor = \boxed{15} \end{aligned}$

Moderator note:

Right, this is a standard geometric progression sum approach.

Bonus question : With your solution, you need to know the value of $\log_{10} 2$ to at least $n$ significant figures. What is this value of $n$ ? And with this value of $n$ , find the value $\log_{10} 2$ up to $n$ significant figures without the use of tables and calculators.

Sir, can we actually compute log 2 without calculator or log tables ? If yes, then please share how. I have seen it for $ln$ but not for base 10.

Vishal Yadav - 5 years, 7 months ago

100 followers problem-2

The answer is 15.

1 solution

Moderator note:

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