Question

If a,b,c are three unequal numbers such that \[a,b,c\] are in A.P and $b - a,c - b,a$ are in G.P, then $a:b:c$ is
$A)1:2:3$
$B)2:3:1$
$C)1:3:2$
$D)3:2:1$

Answer 1

Hint: While talking about the A.P and G.P, we need to know about the concept of Arithmetic and Geometric progression.
An arithmetic progression can be represented by $a,(a + d),(a + 2d),(a + 3d),...$where $a$ is the first term and $d$ is a common difference.
A geometric progression can be given by $a,ar,a{r^2},....$ where $a$ is the first term and $r$ is a common ratio.

Complete step-by-step solution:
To find the ratio relation of the A.P and G.P we first try to find the common difference.
Since from the given that we have, \[a,b,c\] are in A.P. to find the common difference of the arithmetic progression we will subtract the second term with the first term, and we will subtract the third term with the second term and the combined values give as the common difference of the A.P.
Since $b$ is the second term and $a$ is the first term, hence we get a common difference $d = b - a$.
Similarly,, $c$ is the third term and $b$ is the second term, hence we get $d = c - b$.
Thus, combined we have $b - a = c - b$ and take it as an equation $(1)$
Now, to find the common difference for the G.P we have $b - a,c - b,a$ are in G.P,
But in G.P we use ${b^2} = ac$ to find the common difference where here $b = c - b,a = b - a,c = a$ substituting into the formula we get ${b^2} = ac \Rightarrow {(c - b)^2} = (b - a)a$ and now apply the equation $(1)$ $b - a = c - b$, then we get ${(c - b)^2} = (b - a)a \Rightarrow {(b - a)^2} = (b - a)a$
Canceling the common terms, we have $(b - a) = a$ and thus we get $b = 2a$
Again, substitute this value in the equation $(1)$ $b - a = c - b$ we get $ \Rightarrow b - a = c - b \Rightarrow 2a - a = c - 2a$
Further solving we get $3a = c$
Thus, the requirement is $a:b:c$ and we got $b = 2a$ and $3a = c$
Hence, we get $a:b:c = a:2a:3a \Rightarrow 1:2:3$
Therefore, the option $A)1:2:3$

Note: The ratio is the comparison of the two or more than two given quantities by the method of division.
The results of this simplification given the number of times a quantity are equal to another number, thus we get $b$ is two-times multiplied with the number $a$ and $c$ is three times multiplied with the number $a$ and therefore in common ratio, we get $1:2:3$ as the relation of the quantities.