Question:
If the first term of a G.P. a1, a2, a3, ... is unity such that 4 a2 + 5 a3 is least, then the common ratio of G.P. is
(a) −2/5
(b) −3/5
(c) 2/5
(d) none of these
Solution:
(a) $-\frac{2}{5}$
If the first term is 1 , then, the G.P. will be $1, r, r^{2}, r^{3}, \ldots$
Now, $5 r^{2}+4 r=5\left(r^{2}+\frac{4}{5} r\right)$
$=5\left(r^{2}+\frac{4}{5} r+\frac{4}{25}-\frac{4}{25}\right)$
$=5\left(r+\frac{2}{5}\right)^{2}-\frac{4}{5}$
This will be the least when $r+\frac{2}{5}=0$, i. e. $r=-\frac{2}{5}$.
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