Question

If $b_1,b_2,b_3,\dots$ forms a G.P. and $b_1=1,$ then the common ratio of the G.P. when $4b_2+5b_3$ is minimum, is

• A. $-\frac{5}{2}$
• B. $\frac{2}{5}$
• C. $-\frac{2}{5}$
• D. $\frac{5}{2}$

Answer 1

The correct option is C $-\frac{2}{5}$

Given : $b_1=1$, assuming the common ratio of the G.P. be $r$,

Now,
$4b_2+5b_3=4r+5r^2=5[r^2+\frac{4r}{5}+\frac{4}{25}]-\frac{4}{5}=5{(r+\frac{2}{5})}^2-\frac{4}{5}$
So, the minimum value occurs at $r=-\frac{2}{5}$