Question

The sum to $2n$ terms, the terms of the series, ${\log }_{3}1-{\log }_{3}3+{\log }_{3}9-{\log }_3\mathrm{27.......}$ is

• A. $\frac{(2n-1)}{n-1}$
• B. $\frac{(n-1)}{(2n-1)}$
• C. $1-n$
• D. $-n$

Answer 1

The correct option is D $-n$

${\log }_{3}1-1{\log }_{3}3+2{\log }_{3}3-3{\log }_{3}3+...$ (to $2n$ terms).

$=0-1+2-3+....=-(1+3+...+(2n-1\left)\right)+(0+2+4+...+(2n\left)\right)=-n^2+\frac{n}{2}\left(\right(n-1\left)2\right)=-n$