Question

If the first of a G.P. $a_1,a_2,a_3,.......$ is unity such that $4a_2+5a_3$ is least, then the common ratio of G.P. is

• A. $-\frac{2}{5}$
• B. $-\frac{3}{5}$
• C. $-\frac{2}{5}$
• D. none of these

Answer 1

The correct option is A

$-\frac{2}{5}$

(a) $-\frac{2}{5}$

If the first term is 1, then, the G.P. will be 1, $r,r^2,r^3,.....$

Now, $5r^2+4r=5(r^2+\frac{4}{5}r)$

$=5(r^2+\frac{4}{5}r+\frac{4}{25}-\frac{4}{25})$

$=5{(r+\frac{2}{5})}^2-\frac{4}{5}$

This will be the least when $r+\frac{2}{5}=0$

i.e., $r=-\frac{2}{5}$