Question

If \[a,b,c\] are in A.P. and \[a,c - b,b - a\] are in G.P. \[(a \ne b \ne c)\], then \[a:b:c\] is
A. \[1:3:5\]
B. \[1:2:4\]
C. \[1:2:3\]
D. None of these

Answer 1

Hint
A series of terms known as an arithmetic progression (AP) have identical differences. In a geometric progression (GP), the common ratio is multiplied by the previous term to produce each succeeding term. A sort of sequence known as geometric progression (GP) is one in which each following phrase is created by multiplying each preceding term by a fixed number, or "common ratio." This progression is sometimes referred to as a pattern-following geometric sequence of numbers.
A geometric progression is a series where each element is created by multiplying the one before it by a fixed factor.
Formula used:
The arithmetic progression is \[a,b,c\]
\[(b - a) = (c - b)\]
The geometric progression is \[a,b,c\]
\[{b^2} = ac\].
Complete step-by-step solution
The arithmetic progression is \[a,b,c\]
\[(b - a) = (c - b)\] --(1)
The geometric progression is \[a,c-b,b-a\]
\[{(c - b)^2} = (b - a)a\] --(2)
From the equations (1) and (2)
\[ = > {(c - b)^2} = (b - a)a\]
\[ = > {(b - a)^2} = (b - a)a\]
\[ = > (b - a) = a\]
Then, the equation becomes
\[2a = b\]
Substitute the value of b in the equation (1)
\[2a - a = c - 2a\]
\[c = 3a\]
Hence, the equation becomes \[a:b:c = 1:2:3\]
Therefore, the correct option is C.

Note
A series of terms is referred to as a geometric progression if each next term is produced by multiplying each previous term by a fixed amount. (GP), whereas the common ratio is the name given to the constant value. If the ratio of any two subsequent terms is consistently the same, a series of numbers is referred to as a geometric progression. A geometric sequence is a set of numbers where each subsequent element is obtained by multiplying the one before it by a known constant.