Question

The sum of $n$ terms of the sequence $\log a,\log ar,\log a{r}^2,...$ is

• A. $\frac{n}{2}\log a^2{r}^{n-1}$
• B. $n\log a^2{r}^{n-1}$
• C. $\frac{3n}{2}\log a^2{r}^{n-1}$
• D. None of these

Answer 1

The correct option is A $\frac{n}{2}\log a^2{r}^{n-1}$

The given sequence can be expressed as $\log a,(\log a+\log r),(\log a+2\log r),...$

which is clearly an $A.P.$ whose first term is $\log a$ and common difference is $\log r$

The nth term $=\log a+(n-1)\log r$

Since sum to $n$ terms $=\frac{n}{2}(a_1+a_n)$

$\therefore S_n=\frac{n}{2}[\log a+\log a+(n-1)\log r]$

$=\frac{n}{2}[2\log a+(n-1)\log r]=\frac{n}{2}\log a^2{r}^{n-1}$