The first term of the geometric progression (G.P) $b_1, \; b_2, \; b_3, \cdots$ is unity. For what value of the common ratio of the progression is $4 b_2 + 5 b_3$ at a minimum?
First term of the G.P $= b_1 = 1$ $\;\;\;$ (Given)
Let the required common ratio $= r$
$n^{th}$ term of G.P $= b_n = b_1 \cdot r^{n - 1}$
$\therefore \;$ Second term of G.P $= b_2 = b_1 \cdot r = r$
Third term of G.P $= b_3 = b_1 \cdot r^2 = r^2$
Let $\;$ $S = 4 b_2 + 5 b_3$
i.e. $\;$ $S = 4 r + 5 r^2$
For $S$ to be a minimum, $\dfrac{dS}{dr} = 0$
i.e. $\;$ $\dfrac{dS}{dr} = 4 + 10 r = 0$
i.e. $\;$ $r = \dfrac{-4}{10} = \dfrac{-2}{5}$
$\therefore \;$ $S$ is a minimum when $r = \dfrac{-2}{5}$