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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
Options
−2/5
−3/5
2/5
none of these
Solution
− \[\frac{2}{5}\] If the first term is 1, then, the G.P. will be\[1, r, r^2 , r^3 , . . .\]
\[\text{ Now }, 5 r^2 + 4r = 5\left( r^2 + \frac{4}{5}r \right)\]
\[ = 5\left( r^2 + \frac{4}{5}r + \frac{4}{25} - \frac{4}{25} \right)\]
\[ = 5 \left( r + \frac{2}{5} \right)^2 - \frac{4}{5}\]
\[\text{ This will be the least when } r + \frac{2}{5} = 0, i . e . r = - \frac{2}{5} .\]
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