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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

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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

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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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Chapter 20: Geometric Progression - Exercise 20.5 [Page 46]

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R.D. Sharma Mathematics [English] Class 11
Chapter 20 Geometric Progression
Exercise 20.5 | Q 9.2 | Page 46

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