Question

If x is the arithmetic mean of y and z and the two geometric means between y and z are $G_1\mathrm{and}{G}_2$, then $G_1^3+G_2^3=$.

• A. $\mathrm{xyz}$
• B. $\frac{1}{\mathrm{xyz}}$
• C. $2\mathrm{xyz}$
• D. $\frac{2}{\mathrm{xyz}}$

Answer 1

The correct option is C $2\mathrm{xyz}$

It is given that x is the arithmetic mean of y and z.
$\therefore x=\frac{y+z}{2}\Rightarrow 2x=y+z$
Since, $G_1\mathrm{and}{G}_2$ are two geometric means between y and z, so $y,G_1,G_2,z$ are in a GP with common ratio $r={\left(z/y\right)}^{\frac{1}{3}}.G_1=\mathrm{yr}\Rightarrow G_1^3=y^3{r}^3=y^3{\left({\left(\frac{z}{y}\right)}^{\frac{1}{3}}\right)}^3=y^{2}z{G}_2={\mathrm{yr}}^2\Rightarrow G_2^3=y^3{r}^6=y^3{\left({\left(\frac{z}{y}\right)}^{\frac{1}{3}}\right)}^6={\mathrm{yz}}^2$
$G_1^3+G_2^3=y^{2}z+{\mathrm{yz}}^2=\mathrm{yz}(y+z)=\mathrm{yz}\times 2x=2\mathrm{xyz}$