-45+4252.2+4353.3+4454.4+……∞=

-45+12452+13453+...... =log 1-45=log 15 =-log 5

-45+4252.2+4353.3+4454.4+……∞=

-45+12452+13453+...... =log 1-45=log 15 =-log 5

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$- (\frac{4}{5} + \frac{4^{2}}{5^{2} . 2} + \frac{4^{3}}{5^{3} . 3} + \frac{4^{4}}{5^{4} . 4} + \dots \dots \infty) =$

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$- \log 5$

$\log \frac{9}{5}$

$\log (\frac{5}{4})$

$\log \frac{4}{5}$

answer is D.

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Detailed Solution

$\begin{array}{rcl} - [\frac{4}{5} + \frac{1}{2} {(\frac{4}{5})}^{2} + \frac{1}{3} {(\frac{4}{5})}^{3} + . . . . . .] \\ = & \log (1 - \frac{4}{5}) = \log \frac{1}{5} \\ = & - \log 5 \end{array}$

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