Question

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

• A. $\frac{n}{2}\log\left(\frac{a^n}{b^n}\right)$
• B. $\frac{n}{2}\log\left(\frac{a^{n+1}}{b^n}\right)$
• C. $\frac{n}{2}\log\left(\frac{a^{n+1}}{b^{n+1}}\right)$
• D. $\frac{n}{2}\log\left(\frac{a^{n+1}}{b^{n-1}}\right)$

Answer 1

Answer: (D)

$\sum_{r=1}^{n} log\left(\frac{ar}{b^{r-1}}\right)$

$ = log\left(a\right) +log\left(\frac{a^{2}}{b}\right) +log\left(\frac{a^{3}}{b^{2}}\right) + .....+log\left(\frac{a^{n}}{b^{n-1}}\right) $

$= log\left(\frac{a.a^{2}.a^{3}.....a^{n}}{b.b^{2}......b^{n-1}}\right)$

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

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

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