Login/Sign Up
Home » If a, b, c and d are in G.P. show that (a2 + b2 + c2)(b2 + c2 + d2) = (ab + bc + cd)2.
If a, b, c and d are in G.P. show that (a2 + b2 + c2)(b2 + c2 + d2) = (ab + bc + cd)2.
CBSE | Class 11 | Exercise 9.3 | Maths | NCERT | Sequences and Series
If a, b, c and d are in G.P. show that (a2 + b2 + c2)(b2 + c2 + d2) = (ab + bc + cd)2.

Given,

    \[~a,~b,~c,~d~\]

are in G.P.

Thus, we have

    \[\begin{array}{*{35}{l}} bc~=~ad~\text{ }\ldots \text{ }\left( 1 \right) \\ {{b}^{2}}~=~ac~~\ldots \text{ }\left( 2 \right) \\ {{c}^{2}}~=~bd~\text{ }\ldots \text{ }\left( 3 \right) \\ \end{array}\]

Taking the

    \[R.H.S.\]

we have

=R.H.S.

    \[\begin{array}{*{35}{l}} =\text{ }{{\left( ab~+~bc~+~cd \right)}^{2}} \\ =\text{ }{{\left( ab~+~ad~+~cd \right)}^{2}}~\text{ }\left[ Using\text{ }\left( 1 \right) \right] \\ =\text{ }{{\left[ ab~+~d~\left( a~+~c \right) \right]}^{2}} \\ =~{{a}^{2}}{{b}^{2}}~+\text{ }2abd~\left( a~+~c \right)\text{ }+~{{d}^{2}}~{{\left( a~+~c \right)}^{2}} \\ =~{{a}^{2}}{{b}^{2}}~+2{{a}^{2}}bd~+\text{ }2acbd~+~{{d}^{2}}\left( {{a}^{2}}~+\text{ }2ac~+~{{c}^{2}} \right) \\ =~{{a}^{2}}{{b}^{2}}~+\text{ }2{{a}^{2}}{{c}^{2}}~+\text{ }2{{b}^{2}}{{c}^{2}}~+~{{d}^{2}}{{a}^{2}}~+\text{ }2{{d}^{2}}{{b}^{2}}~+~{{d}^{2}}{{c}^{2}}~\text{ }\left[ Using\text{ }\left( 1 \right)\text{ }and\text{ }\left( 2 \right) \right] \\ =~{{a}^{2}}{{b}^{2}}~+~{{a}^{2}}{{c}^{2}}~+~{{a}^{2}}{{c}^{2}}~+~{{b}^{2}}{{c}^{2~}}+~{{b}^{2}}{{c}^{2}}~+~{{d}^{2}}{{a}^{2}}~+~{{d}^{2}}{{b}^{2}}~+~{{d}^{2}}{{b}^{2}}~+~{{d}^{2}}{{c}^{2}} \\ =~{{a}^{2}}{{b}^{2}}~+~{{a}^{2}}{{c}^{2}}~+~{{a}^{2}}{{d}^{2~}}+~{{b}^{2~}}\times ~{{b}^{2}}~+~{{b}^{2}}{{c}^{2}}~+~{{b}^{2}}{{d}^{2}}~+~{{c}^{2}}{{b}^{2}}~+~{{c}^{2~}}\times ~{{c}^{2}}~+~{{c}^{2}}{{d}^{2}} \\ \end{array}\]

[Using

    \[\left( 2 \right)\text{ }and\text{ }\left( 3 \right)\]

and reworking terms]

    \[\begin{array}{*{35}{l}} =~{{a}^{2}}\left( {{b}^{2}}~+~{{c}^{2}}~+~{{d}^{2}} \right)\text{ }+~{{b}^{2}}~\left( {{b}^{2}}~+~{{c}^{2}}~+~{{d}^{2}} \right)\text{ }+~{{c}^{2}}~\left( {{b}^{2}}+~{{c}^{2}}~+~{{d}^{2}} \right) \\ =\text{ }\left( {{a}^{2}}~+~{{b}^{2}}~+~{{c}^{2}} \right)\text{ }\left( {{b}^{2}}~+~{{c}^{2}}~+~{{d}^{2}} \right) \\ \end{array}\]

    \[=\text{ }L.H.S.\]

Accordingly,

    \[L.H.S.\text{ }=\text{ }R.H.S.\]

Accordingly,

    \[({{a}^{2}}~+\text{ }{{b}^{2}}~+\text{ }{{c}^{2}})({{b}^{2}}~+\text{ }{{c}^{2}}~+\text{ }{{d}^{2}})\text{ }=\text{ }{{\left( ab\text{ }+\text{ }bc\text{ }+\text{ }cd \right)}^{2}}\]

i

More Material