Question

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 ?

• A. $xyz$
• B. $x{y}^{2}z$
• C. $xy{z}^2$
• D. $2xyZ$

Answer 1

Answer: (D)

Since $x$ is $A.M$
$\Rightarrow x=\frac{y+z}{2}$,
$\Rightarrow 2x=y+z$
and $y,g_1,g_2,z\dots$. 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)
$g_1^3+g_2^3=y{g}_1{g}_2+z{g}_1{g}_2=g_1{g}_2(y+z)$
$=yz\cdot 2x=2xyz$