The value of sumlimits_{r = 1}^n {log left( { | vedclass

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(C) The given series is $S = \sum_{r=1}^n \log \left( \frac{a^r}{b^{r-1}} \right) = \log a + \log \left( \frac{a^2}{b} \right) + \log \left( \frac{a^3

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$\frac{n}{2}\log \left( {\frac{{{a^{n + 1}}}}{{{b^{n - 1}}}}} \right)$

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(C) The given series is $S = \sum_{r=1}^n \log \left( \frac{a^r}{b^{r-1}} \right) = \log a + \log \left( \frac{a^2}{b} \right) + \log \left( \frac{a^3}{b^2} \right) + \dots + \log \left( \frac{a^n}{b^{n-1}} \right)$.This is an arithmetic progression $(A.P.)$ where the first term $a_1 = \log a$ and the $n$-th term $a_n = \log \left( \frac{a^n}{b^{n-1}} \right)$.The sum of $n$ terms of an $A.P.$ is given by $S_n = \frac{n}{2} (a_1 + a_n)$.Substituting the values,we get $S_n = \frac{n}{2} \left[ \log a + \log \left( \frac{a^n}{b^{n-1}} \right) \right]$.Using the property $\log x + \log y = \log(xy)$,we have $S_n = \frac{n}{2} \log \left( a \cdot \frac{a^n}{b^{n-1}} \right) = \frac{n}{2} \log \left( \frac{a^{n+1}}{b^{n-1}} \right)$.

The value of $\sum\limits_{r = 1}^n {\log \left( {\frac{{{a^r}}}{{{b^{r - 1}}}}} \right)} $ is

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