First slide

Arithmetic progression

Question

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

Moderate

A

$\frac{n}{2} \log a^{2} r^{n - 1}$
B

$nlog a^{2} r^{n - 1}$
C

$\frac{3 n}{2} \log a^{2} r^{n - 1}$
D

None of these

Solution

The given sequence can be expressed as
$\log a, (\log a + \log r), (\log a + 2 \log r) \dots$
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 $s = \frac{n}{2} (a_{1} + a_{n})$
$∴ S_{n} = \frac{n}{2} [\log a + \log a + (n - 1) \log r]$
$= \frac{n}{2} [2 \log a + (n - 1) \log r] = \frac{n}{2} \log a^{2} r^{n - 1}$

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