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Question

If $$x, 2y, 3z$$ are in AP, where the distinct numbers $$x, y, z$$ are in GP, then the common ratio of the GP is:


A
3
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B
13
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C
2
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D
12
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Solution

The correct option is D $$\displaystyle \frac{1}{3}$$
Given $$ 4y = x+3z $$ _____(1)
and $$ y^{2}=xz$$
Let common ratio be $$r$$.
$$\therefore \displaystyle \frac{y}{x}= r $$ and $$ \displaystyle\frac{z}{x}=r^{2} $$
Dividing (1) by x,  $$4\displaystyle\frac{y}{x} = 1+ 3\displaystyle\frac{z}{x} $$
$$\Rightarrow 4r=1+3r^{2} $$
$$\Rightarrow 3r^{2}-4r+1=0 $$
$$\Rightarrow (3r-1)(r-1)=0 $$
$$\Rightarrow r= \displaystyle\frac{1}{3}, 1$$
But $$r \neq 1  \quad (\because x,y,z $$ are distinct) 
$$\therefore r=\displaystyle\frac{1}{3} $$

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