Question

If $A=1+r^a+r^{2a}+r^{3a}+........\infty$ and $B=1+r^b+r^{2b}+r^{3b}+........\infty$ then $\frac{a}{b}$ is equal to

• A. $log_{1_B}(1 - A)$
• B. $log_{\frac{B-1}{B}}\left(\frac{A-1}{A}\right)$
• C. $ log_B A$
• D. none of these

Answer 1

Answer: (B)

$A = \frac{1}{1-r^{a}}$
$\Rightarrow 1-r^{a} = \frac{1}{A} $
$ \Rightarrow r^{a} = 1-\frac{1}{A} =\frac{ A-1}{A} $
$B= \frac{1}{1-r^{b}} $
$ \Rightarrow r^{b} = 1 -\frac{1}{B} = \frac{B-1}{B} $
$\therefore a \,log \,r = log_{r}\, a = log \left(\frac{A-1}{A}\right), b \,log\, r$
$ = log_{a}^{b} = log\left(\frac{B-1}{B} \right)$
$\therefore \frac{a}{b} = \frac{log\left(\frac{A-1}{A}\right)}{log\left(\frac{B-1}{B}\right) } = log_{\frac{B-1}{B} }\left(\frac{A-1}{A}\right)$