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# If $x,2y,3z$ are in AP, where the distinct numbers $x,y,z$ are in GP, then the common ratio is:A. $3$B. $\dfrac{1}{3}$C. $2$D. $\dfrac{1}{2}$

Last updated date: 09th Aug 2024
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Hint:We are given that certain series are in AP and some in GP. Writing the formulas for the AP and GP. Which will form two equations and by using the formula for common ratio which will form a quadratic equation. By solving the quadratic equation, we can find or answer.

Complete step by step solution:
Firstly, learning the meaning of AP and GP. AP full form is arithmetic progression. AP means sequence of numbers having the same difference between the consecutive numbers. It can be written as $a,a + d,a + 2d, \ldots,\ldots$ where d is the common difference.
It can be written as $2\left( {a + d} \right) = a + \left( {a + 2d} \right)$
Whereas GP full form is geometric progression which means if in a sequence of terms, each succeeding term is generated by multiplying each preceding term with the constant value and that constant value is known as the common ratio. It can be written in the form of $a,ar,a{r^2},a{r^3} \ldots \ldots$ where r is the common ratio.
It can be written as ${\left( {ar} \right)^2} = \left( a \right)\left( {a{r^2}} \right)$
According to the question, we are provided that $x,2y,3z$are in AP. So, these numbers can be written as
$2\left( {2y} \right) = x + 3z$ $\ldots \left( 1 \right)$
And $x,y,z$are in GP. So, these can be written as ${y^2} = xz$ $\ldots \left( 2 \right)$
In the question, we have to find the common ratio. So, let the common ratio be $r$ and it should be equal to the $r = \dfrac{y}{x}$ and ${r^2} = \dfrac{z}{x}$
In order to find r, dividing the equation $\left( 1 \right)$ by x
$4\dfrac{y}{x} = 1 + 3\dfrac{z}{x}$ and now putting the value of r,
$4r = 1 + 3{r^2}$ or $3{r^2} - 4r + 1 = 0$
$3{r^2} - 3r - r + 1 = 0 \\ 3r\left( {r - 1} \right) - 1\left( {r - 1} \right) = 0 \\ \left( {3r - 1} \right)\left( {r - 1} \right) = 0 \\$
Hence, $r = 1,\dfrac{1}{3}$
But $r = 1$ is neglected as $x,y,z$ are distinct.
Hence, the common ratio is $\dfrac{1}{3}$. So, the correct option is B.