The sum of first four terms of a G.P is 6512 and the sum of their reciprocals is 6518. If the product of first three terms of G.P is 1 and third term is α then 2α is _________

a+ar+ar2+ar3=65121a+1ar+1ar2+1ar3=6518⇒a2r3=32 Given a×ar×ar2=1⇒a=23⇒r=32∴RV=3

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The sum of first four terms of a G.P is $\frac{65}{12}$ and the sum of their reciprocals is $\frac{65}{18} .$ If the product of first three terms of G.P is 1 and third term is $α$ then $2 α$ is _________

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$a + a r + a r^{2} + a r^{3} = \frac{65}{12}$

$\frac{1}{a} + \frac{1}{a r} + \frac{1}{a r^{2}} + \frac{1}{a r^{3}} = \frac{65}{18}$

$\Rightarrow a^{2} r^{3} = \frac{3}{2}$

Given $a \times a r \times a r^{2} = 1$

$\Rightarrow a = \frac{2}{3}$

$\Rightarrow r = \frac{3}{2}$

$∴ R V = 3$

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