Question

The first term of an A.P is $\log a$ and the second term is $\log b$ . Express the sum to "n" terms as a logarithm.

• A. $=\log {\left(\frac{b^{n-1}}{a^{n+3}}\right)}^{n/2}$
• B. $=\log {\left(\frac{b^{n-1}}{a^{n-2}}\right)}^{n/2}$
• C. $=\log {\left(\frac{b^{n-1}}{a^{n-3}}\right)}^{n/2}$
• D. $=\log {\left(\frac{b^{n-1}}{a^{n-3}}\right)}^{n/3}$

Answer 1

The correct option is C $=\log {\left(\frac{b^{n-1}}{a^{n-3}}\right)}^{n/2}$

From the given things we can write,

a=$\log a$

$d=\log b-\log a$ = $\log \frac{b}{a}$

$S_n=\frac{n}{2}[2a+(n-1\left)d\right]$

$=\frac{n}{2}[2\log a+(n-1\left)\log \frac{b}{a}\right]$

$=\frac{n}{2}[\log a^2+n\log \frac{b}{a}-\log \frac{b}{a}]$

$=\frac{n}{2}[\log a^2+\log {\left[\frac{b}{a}\right]}^n-\log \frac{b}{a}]$

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

$=\frac{n}{2}\left[\log \left(\frac{b^{n-1}}{a^{n-3}}\right)\right]\dots (\log a-\log b=\log \frac{a}{b})$

$S_n=\log {\left(\frac{b^{n-1}}{a^{n-3}}\right)}^{n/2}\dots (n{\log }_{y}x={\log }_y{x}^n)$