If the first term of a G.P., $a_{1}, a_{2}, a_{3}, \dots$ is unity, then the value of $4 a_{2} + 5 a_{3}$ will be minimum when the common ratio is

\frac{1}{3}

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- \frac{1}{3}

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\frac{2}{5}

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- \frac{2}{5}

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Solution

The correct option is D $- \frac{2}{5}$
Let $r$ be the common ratio.
$a_{1} = 1,$ then $a_{2} = r, a_{3} = r^{2}$
$∴ 4 a_{2} + 5 a_{3} = 4 r + 5 r^{2}$ is minimum when $r = - \frac{4}{2 \times 5} = - \frac{2}{5}$
$(a > 0, a x^{2} + b x + c is minimum at x = - \frac{b}{2 a})$

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