Question

The sum of $n$terms of the sequence $\log a,\log ar,\log a{r^2},...$ is:
A.$\dfrac{n}{2}\log {a^2}{r^{n - 1}}$
B.$n\log {a^2}{r^{n - 1}}$
C.$\dfrac{{3n}}{2}\log {a^2}{r^{n - 1}}$
D.None of these

Answer 1

Hint: We first simplify the given sequence and then find its first term and common difference. To find the sum of $n$terms of the given sequence, use the formula , ${S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$. To get the final answer, use the properties of log to simplify the answer.

Complete step by step answer:

The first term of the sequence is $\log a$
We will first simplify the given sequence.
We know that, $\log \left( {xy} \right) = \log x + \log y$
Therefore, we write the given sequence as, $\log a,\log a + \log r,\log a + \log {r^2},...$
Also, $\log \left( {{x^m}} \right) = m\log x$
Hence, we have, $\log a,\log a + \log r,\log a + 2\log r,...$
Now, we can observe that the given sequence is an AP as $\log r$ is added to each term.
Now, we will find the common difference by subtracting the second tern from the first one.
Hence, we get common difference $d$ as,
$d = \log a + \log r - \log a \\\ d = \log r \\\$
We have to find the sum of $n$terms of the given sequence.
We know that the sum of $n$ terms of a sequence is $\dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$, where $a$ is the first term and $d$ is the common difference.
Substitute the values of the first term and the common difference to find the sum of $n$terms of the sequence.
Hence, sum of $n$ terms is
$\dfrac{n}{2}\left( {2\left( {\log a} \right) + \left( {n - 1} \right)\log r} \right)$
We can simplify the above expression using the properties of log, $\log \left( {{x^m}} \right) = m\log x$
$\Rightarrow$ $\dfrac{n}{2}\left( {\log {a^2} + \log {r^{n - 1}}} \right)$
Hence, option A is correct.

Note: We have to use properties of log to simplify the given sequence such as, $\log \left( {xy} \right) = \log x + \log y$ and $\log \left( {{x^m}} \right) = m\log x$. Many students make mistakes by considering the given sequence as GP. It is after applying the properties of log, we will observe that the given sequence is an AP.