Question

If we have the expression \[\dfrac{{\left( {{\text{ }}a{\text{ }} + {\text{ }}bx{\text{ }}} \right){\text{ }}}}{{\left( {{\text{ }}a{\text{ }} - {\text{ }}bx{\text{ }}} \right)}} = \dfrac{{\left( {{\text{ }}b{\text{ }} + {\text{ }}cx{\text{ }}} \right)}}{{\left( {{\text{ }}b{\text{ }} - {\text{ }}cx{\text{ }}} \right)}}{\text{ }} = {\text{ }}\dfrac{{\left( {{\text{ }}c{\text{ }} + {\text{ }}dx{\text{ }}} \right){\text{ }}}}{{\left( {{\text{ }}c{\text{ }} - {\text{ }}dx{\text{ }}} \right)}},\]\[\left( {x{\text{ }} \ne {\text{ }}0} \right)\]then \[a,{\text{ }}b,{\text{ }}c,{\text{ }}d\]are in
\[\left( 1 \right)\]\[AP\]
\[\left( 2 \right)\]\[GP\]
\[\left( 3 \right)\]\[HP\]
\[\left( 4 \right)\]\[None{\text{ }}of{\text{ }}these\]

Answer 1

Hint: We have to find the relation between \[a{\text{ }},{\text{ }}b{\text{ }},{\text{ }}c\]and\[d\]. We have to compute the relation between these four variables using the given expression . We should have the knowledge of the concept of arithmetic progression\[\left( {A.P.} \right)\], geometric progression\[\left( {G.P.} \right)\], harmonic progression\[\left( {H.P.} \right)\]. we should also have the concept of the mean of the progressions . Solving the expression we can compute the relation between \[a{\text{ }},{\text{ }}b{\text{ }},{\text{ }}c\] and \[d\].

Complete step-by-step solution:
For the terms of a given series to be in G.P. the common ratio between the terms of the series should be the same for all the two consecutive terms of the series . The ratio of the second term to the first term of the given series should be the same as that of the ratio of the third term to the second term of the given series .
Given :
\[\dfrac{{\left( {{\text{ }}a{\text{ }} + {\text{ }}bx{\text{ }}} \right){\text{ }}}}{{\left( {{\text{ }}a{\text{ }} - {\text{ }}bx{\text{ }}} \right)}} = \dfrac{{\left( {{\text{ }}b{\text{ }} + {\text{ }}cx{\text{ }}} \right)}}{{\left( {{\text{ }}b{\text{ }} - {\text{ }}cx{\text{ }}} \right)}}{\text{ }} = {\text{ }}\dfrac{{\left( {{\text{ }}c{\text{ }} + {\text{ }}dx{\text{ }}} \right){\text{ }}}}{{\left( {{\text{ }}c{\text{ }} - {\text{ }}dx{\text{ }}} \right)}}\]
Taking two terms at a time . Let us take the first two terms of the expression , we get
\[\dfrac{{\left( {{\text{ }}a{\text{ }} + {\text{ }}bx{\text{ }}} \right){\text{ }}}}{{\left( {{\text{ }}a{\text{ }} - {\text{ }}bx{\text{ }}} \right)}} = \dfrac{{\left( {{\text{ }}b{\text{ }} + {\text{ }}cx{\text{ }}} \right)}}{{\left( {{\text{ }}b{\text{ }} - {\text{ }}cx{\text{ }}} \right)}}{\text{ }}\]
Cross multiplying the terms , we get
\[\left( {{\text{ }}a{\text{ }} + {\text{ }}bx{\text{ }}} \right){\text{ }} \times {\text{ }}\left( {{\text{ }}b{\text{ }} - {\text{ }}cx{\text{ }}} \right){\text{ }} = {\text{ }}\left( {{\text{ }}b{\text{ }} + {\text{ }}cx{\text{ }}} \right){\text{ }} \times {\text{ }}\left( {{\text{ }}a{\text{ }} - {\text{ }}bx{\text{ }}} \right)\]
Expanding the terms , we get
$ab + {b^2} \times x - acx - bc \times {x^2} = ba + acx - {b^2} \times x - bc \times {x^2}$
After cancelling the like terms , we get
${b^2} \times x - acx = acx - {b^2} \times x$
On simplifying , we get
${b^2} \times x = acx$
${b^2} = ac$———(1)
From \[\left( 1 \right)\]we conclude that \[a{\text{ }},{\text{ }}b\]and $c$ are in G.P.
Similarly taking second and third term , we get
\[\dfrac{{\left( {{\text{ }}b{\text{ }} + {\text{ }}cx{\text{ }}} \right)}}{{\left( {{\text{ }}b{\text{ }} - {\text{ }}cx{\text{ }}} \right)}}{\text{ }} = {\text{ }}\dfrac{{\left( {{\text{ }}c{\text{ }} + {\text{ }}dx{\text{ }}} \right){\text{ }}}}{{\left( {{\text{ }}c{\text{ }} - {\text{ }}dx{\text{ }}} \right)}}\]
Cross multiplying the terms , we get
\[\left( {{\text{ }}b{\text{ }} + {\text{ }}cx{\text{ }}} \right){\text{ }} \times {\text{ }}\left( {{\text{ }}c{\text{ }} - {\text{ }}dx{\text{ }}} \right){\text{ }} = {\text{ }}\left( {{\text{ }}c{\text{ }} + {\text{ }}dx{\text{ }}} \right){\text{ }} \times {\text{ }}\left( {{\text{ }}b{\text{ }} - {\text{ }}cx{\text{ }}} \right)\]
Expanding the terms , we get
$bc + {c^2} \times x - bdx - cd \times {x^2} = cb + bdx - {c^2} \times x - cd \times {x^2}$
After cancelling the like terms , we get
${c^2} \times x - bdx = bdx - {c^2} \times x$
On simplifying , we get
${c^2} \times x = bdx$
${c^2} = bd$———(2)
From \[\left( 2 \right)\]we conclude that \[b{\text{ }},{\text{ }}c\]and $d$ are in (G.P.) .
From\[\left( 1 \right)\], we get
\[{\text{ }}\dfrac{b}{c} = {\text{ }}\dfrac{a}{b}\]———(3)
From\[\left( 2 \right)\], we get
\[\dfrac{b}{c}{\text{ }} = \dfrac{c}{d}\]———(4)
From \[\left( 3 \right)\]and\[\left( 4 \right)\], we get
a / b = c / d
From the relations , we get
\[\dfrac{a}{b}{\text{ }} = {\text{ }}\dfrac{b}{c}{\text{ }} = \;\dfrac{c}{d}\]
Thus \[a{\text{ }},{\text{ }}b{\text{ }},{\text{ }}c\]and $d$ are in a geometric progression \[\left( {G.P.} \right)\]
Hence , the correct option is\[\left( 2 \right)\].

Note: For the terms of a given series to be in G.P. the common ratio between the terms of the series should be the same for all the two consecutive terms of the series . The ratio of the second term to the first term of the given series should be the same as that of the ratio of the third term to the second term of the given series .
The formula of mean of the three progression is given as :
\[\left( 1 \right){\text{ }}A.P.\]
Arithmetic mean \[ = {\text{ }}\dfrac{{\left( {{\text{ }}a{\text{ }} + {\text{ }}b{\text{ }}} \right)}}{2}\]
\[\left( 2 \right){\text{ }}G.P.\]
Geometric mean $ = \sqrt {(b \times c)} $
\[\left( 3 \right){\text{ }}H.P.\]
Harmonic mean = \[\dfrac{{2{\text{ }}ab}}{{\left( {{\text{ }}a{\text{ }} + {\text{ }}b{\text{ }}} \right)}}\].