First slide

Sigma operation series

Question

The sum of the series $S = \sum_{r - 1}^{n} \log (\frac{a^{r + 1}}{b^{r - 1}})$ is

Moderate

A

$\frac{n}{2} \log (\frac{a^{n - 1}}{b^{n}})$
B

$\frac{n}{2} \log (\frac{a^{n + 3}}{b^{n - 1}})$
C

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

none of these

Solution

$S = \sum_{r = 1}^{n} [(r + 1) \log a - (r - 1) \log b]$ .
$= \frac{n}{2} [(2 + n + 1) \log a - (0 + n - 1) \log b]$
$= \frac{n}{2} \log (\frac{a^{n + 3}}{b^{n - 1}})$

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