Question

Find the sum of $n$ terms of the series:
$\log a+\log \frac{a^2}{b}+\log \frac{a^3}{b^2}+\log \frac{a^4}{b^3}+\dots \dots \dots \dots$ $n$ terms.

Answer 1

$Firstterm=loga$

$Secondterm=log\left(\frac{a^2}{b}\right)=2loga-logb$

$Thirdterm=log\left(\frac{a^3}{b^2}\right)=3loga-2logb$

$\therefore a_n=nloga-(n-1)logb$

$\therefore S_n=\sum a_n$

$=\sum nloga-(n-1)logb$

$=\left(\frac{n(n+1)}{2}\right)loga-\left(\frac{(n-1)n}{2}\right)logb$

$=\left(\frac{n}{2}\right)\left[\right(n+1)loga-(n-1\left)logb\right]$

$=\left(\frac{n}{2}\right)[nloga+loga-nlogb+logb]$

$=\left(\frac{n}{2}\right)[loga+logb+n(loga-logb\left)\right]$

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