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Question
If a, b, c, d are in G.P., prove that:
(a + b + c + d)2 = (a + b)2 + 2 (b + c)2 + (c + d)2
Solution
a, b, c and d are in G.P.
\[\therefore b^2 = ac\]
\[bc = ad\]
\[ c^2 = bd\] .......(1)
\[\text { LHS }= \left( a + b + c + d \right)^2 \]
\[ = \left( a + b \right)^2 + 2\left( a + b \right)\left( c + d \right) + \left( c + d \right)^2 \]
\[ = \left( a + b \right)^2 + 2\left( ac + ad + bc + bd \right) + \left( c + d \right)^2 \]
\[ = \left( a + b \right)^2 + 2\left( b^2 + bc + bc + c^2 \right) + \left( c + d \right)^2 \left[ \text { Using } (1) \right]\]
\[ = \left( a + b \right)^2 + 2 \left( b + c \right)^2 + \left( c + d \right)^2 = \text { RHS }\]
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