Thank you for reporting, we will resolve it shortly
Q.
Let $x$ be one A.M and $g_{1}$ and $g_{2}$ be two G.Ms between $y$ and $z$. What is $g _{1}^{3}+ g _{2}^{3}$ equal to ?
Sequences and Series
Solution:
Since $x$ is $A . M$
$ \Rightarrow x=\frac{y+z}{2}$,
$\Rightarrow 2 x=y+z$
and $y , g _{1}, g _{2}, z \ldots$. are in G.P.
$\Rightarrow \frac{ g _{1}}{ y }=\frac{ g _{2}}{ g _{1}}=\frac{ z }{ g _{2}}$
$\Rightarrow g _{1}^{2}= g _{2} y$
$ \Rightarrow g _{1}^{3}= g _{1} g _{2} y\, \dots(ii)$
Also, $g_{2}^{2}=g_{1} z$
$ \Rightarrow g_{2}^{3}=g_{1} g_{2} z\, \dots(iii)$
$\Rightarrow g _{1}^{2} g _{2}^{2}= g _{1} g _{2} yz$
$ \Rightarrow yz = g _{1} g _{2}\, \dots(iv)$
Adding equations (ii) and (iii)
$g _{1}^{3}+ g _{2}^{3}= yg _{1} g _{2}+ zg _{1} g _{2}= g _{1} g _{2}( y + z )$
$= yz \cdot 2 x =2 xyz$