Question

If \[\frac{{a + bx}}{{a - bx}} = \frac{{b + cx}}{{b - cx}} = \frac{{c + dx}}{{c - dx}}(x \ne 0)\], then \[a,b,c,d\] are in
A. A.P.
B. G.P.
C. H.P.
D. None of these

Answer 1

Hint
The proportions theorem of "componendo and dividendo" allows fast calculations and minimizes the quantity of expansions needed. It's notably useful for determination three-quarter equations or rational equations in mathematical Olympiads, particularly once fractions area unit gift. A proportion is Associate in Nursing equation that sets 2 ratios at constant price.
If each part of the set has Associate in Nursing inverse for a selected operation, the set has the inverse attribute. Associate in Nursing part's inverse may be a completely different element within the set that, once coupled on the correct or left by Associate in Nursing operation, invariably yields the operator. The phrase "when we tend to multiply any variety by zero, the ensuing product is usually a zero" is employed to outline the zero property of multiplication.
Formula used:
\[a,b,c\] in G.P
\[ {b^2} = ac\]
Complete step-by-step solution
The given equation is
\[\frac{{a + bx}}{{a - bx}} = \frac{{b + cx}}{{b - cx}}\]
Cross multiply the equation
\[(a + bx)(b - cx) = (b + cx)(a - bx)\]
\[ = > ab + {b^2}x - acx - bc{x^2} = ab + acx - {b^2}x - bc{x^2}\]
It can be solved as
\[ = > 2{b^2}x = 2acx\]
\[ = > {b^2} = ac\] ---(1)
Another equation given is
\[\frac{{b + cx}}{{b - cx}} = \frac{{c + dx}}{{c - dx}}\]
Cross multiply this equation to get
\[(b + cx)(c - dx) = (c + dx)(b - cx)\]
\[ = > bc + {c^2}x - bdx - cd{x^2} = bc + bdx - {c^2}x - cd{x^2}\]
\[ = > 2{c^2}x = 2bdx\]
\[ = > {c^2} = bd\] ---(2)
\[a,b,c,d\] in G.P from the equations (1) and (2)
Therefore, the correct option is B.
Note
A mathematical sequence known as a geometric progression (GP) is one in which each following phrase is generated by multiplying each preceding term by a fixed integer, or "common ratio." This progression is sometimes referred to as a pattern-following geometric sequence of numbers. In an arithmetic progression (AP), the differences between every two consecutive terms are all the same, whereas in a geometric progression, the ratios of every two successive terms are all the same (GP).