Given,
![\[~a,~b,~c,~d~\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-91e693c93bebfb39eedc78359f9f6994_l3.png "Rendered by QuickLaTeX.com")
are in G.P.
Thus, we have
![\[\begin{array}{*{35}{l}} bc~=~ad~\text{ }\ldots \text{ }\left( 1 \right) \\ {{b}^{2}}~=~ac~~\ldots \text{ }\left( 2 \right) \\ {{c}^{2}}~=~bd~\text{ }\ldots \text{ }\left( 3 \right) \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-cb895d0d1c3e29e2b63fff701278dc79_l3.png "Rendered by QuickLaTeX.com")
Taking the
![\[R.H.S.\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-24d6d3e96b62cabe87dcafc4fc0d38ea_l3.png "Rendered by QuickLaTeX.com")
we have
![\[\begin{array}{*{35}{l}} =\text{ }{{\left( ab~+~bc~+~cd \right)}^{2}} \\ =\text{ }{{\left( ab~+~ad~+~cd \right)}^{2}}~\text{ }\left[ Using\text{ }\left( 1 \right) \right] \\ =\text{ }{{\left[ ab~+~d~\left( a~+~c \right) \right]}^{2}} \\ =~{{a}^{2}}{{b}^{2}}~+\text{ }2abd~\left( a~+~c \right)\text{ }+~{{d}^{2}}~{{\left( a~+~c \right)}^{2}} \\ =~{{a}^{2}}{{b}^{2}}~+2{{a}^{2}}bd~+\text{ }2acbd~+~{{d}^{2}}\left( {{a}^{2}}~+\text{ }2ac~+~{{c}^{2}} \right) \\ =~{{a}^{2}}{{b}^{2}}~+\text{ }2{{a}^{2}}{{c}^{2}}~+\text{ }2{{b}^{2}}{{c}^{2}}~+~{{d}^{2}}{{a}^{2}}~+\text{ }2{{d}^{2}}{{b}^{2}}~+~{{d}^{2}}{{c}^{2}}~\text{ }\left[ Using\text{ }\left( 1 \right)\text{ }and\text{ }\left( 2 \right) \right] \\ =~{{a}^{2}}{{b}^{2}}~+~{{a}^{2}}{{c}^{2}}~+~{{a}^{2}}{{c}^{2}}~+~{{b}^{2}}{{c}^{2~}}+~{{b}^{2}}{{c}^{2}}~+~{{d}^{2}}{{a}^{2}}~+~{{d}^{2}}{{b}^{2}}~+~{{d}^{2}}{{b}^{2}}~+~{{d}^{2}}{{c}^{2}} \\ =~{{a}^{2}}{{b}^{2}}~+~{{a}^{2}}{{c}^{2}}~+~{{a}^{2}}{{d}^{2~}}+~{{b}^{2~}}\times ~{{b}^{2}}~+~{{b}^{2}}{{c}^{2}}~+~{{b}^{2}}{{d}^{2}}~+~{{c}^{2}}{{b}^{2}}~+~{{c}^{2~}}\times ~{{c}^{2}}~+~{{c}^{2}}{{d}^{2}} \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-46e583c1ff4a05b009313431c92e97ab_l3.png "Rendered by QuickLaTeX.com")
[Using
![\[\left( 2 \right)\text{ }and\text{ }\left( 3 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-8a94e5ceedf37f66992a3bca450f381f_l3.png "Rendered by QuickLaTeX.com")
and reworking terms]
![\[\begin{array}{*{35}{l}} =~{{a}^{2}}\left( {{b}^{2}}~+~{{c}^{2}}~+~{{d}^{2}} \right)\text{ }+~{{b}^{2}}~\left( {{b}^{2}}~+~{{c}^{2}}~+~{{d}^{2}} \right)\text{ }+~{{c}^{2}}~\left( {{b}^{2}}+~{{c}^{2}}~+~{{d}^{2}} \right) \\ =\text{ }\left( {{a}^{2}}~+~{{b}^{2}}~+~{{c}^{2}} \right)\text{ }\left( {{b}^{2}}~+~{{c}^{2}}~+~{{d}^{2}} \right) \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-efd2aadd0eac30ac4d4edbd609a3ed8b_l3.png "Rendered by QuickLaTeX.com")
![\[=\text{ }L.H.S.\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-af834ba74590df1aca660299a84cbe96_l3.png "Rendered by QuickLaTeX.com")
Accordingly,
![\[L.H.S.\text{ }=\text{ }R.H.S.\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-40fe35afc3049cb7c7e19f80dca3b0d0_l3.png "Rendered by QuickLaTeX.com")
Accordingly,
![\[({{a}^{2}}~+\text{ }{{b}^{2}}~+\text{ }{{c}^{2}})({{b}^{2}}~+\text{ }{{c}^{2}}~+\text{ }{{d}^{2}})\text{ }=\text{ }{{\left( ab\text{ }+\text{ }bc\text{ }+\text{ }cd \right)}^{2}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-aa5b7be08bb3a7e0cde2a6e6f11d2796_l3.png "Rendered by QuickLaTeX.com")
Given,
are in G.P.
Thus, we have
Taking the
we have
[Using
and reworking terms]
Accordingly,
Accordingly,
