Question

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

Answer 1

$T_n-T_{n-1}=log\frac{a^n}{b^{n-1}}-log\frac{a^{n-1}}{b^{n-2}}$
$=log\frac{a^n}{b^{n-1}}\cdot \frac{b^{n-2}}{a^{n-1}}=log\frac{a}{b}=$constant as it is independent of n. Hence the given series is an A.P. whose A$=loga$ and $D=log\frac{a}{b}$.
$\therefore S=\frac{n}{2}[2loga+(n-1\left)log\frac{a}{b}\right]$
$=\frac{n}{2}[nlog\frac{a}{b}+2loga-(loga-logb\left)\right]$
$=\frac{n}{2}[nlog\frac{a}{b}+logab]$.