Question

The first term of geometric progression $b_1,b_2,b_3,\dots$ is unity. The minimum value of $4b_2+5b_3$ is

• A. $-\frac{2}{5}$
• B. $-\frac{4}{5}$
• C. $-\frac{1}{5}$
• D. $-\frac{8}{5}$

Answer 1

The correct option is B $-\frac{4}{5}$

$b_1=1$

$b_2=1.r=r$

$b_3=1.r^2=r^2$

$S=4b_2+5b_3=4r+5r^2$

$\frac{ds}{dr}=4+10r=0$

$\Rightarrow r=\frac{-2}{5}$

$\frac{d^{2}s}{d{r}^2}=10>0\Rightarrow$ Minima

$\Rightarrow r=\frac{-2}{5}$ gives minimum value

$\Rightarrow S=4b_2+5b_3$

$=4r+5r^2$

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

$=-\frac{8}{5}+\frac{4}{5}$

$=-\frac{4}{5}$.