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If (A − B), (B − C), (C − A) Are in G.P., Then Prove that (A + B + C)2 = 3 (Ab + Bc + Ca) - Mathematics

If (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)

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Solution

\[\left( a - b \right), \left( b - c \right) \text { and  }\left( c - a \right) \text { are in G . P} . \]

\[ \therefore \left( b - c \right)^2 = \left( a - b \right)\left( c - a \right)\]

\[ \Rightarrow b^2 - 2bc + c^2 = ac - bc + ab - a^2 \]

\[ \Rightarrow a^2 + b^2 + c^2 = ab + bc + ca . . . . . . . (i)\]

\[\text{ Now, LHS } = \left( a + b + c \right)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca\]

\[ = ab + bc + ca + 2ab + 2bc + 2ca \left[\text {  Using  }(i) \right]\]

\[ = 3ab + 3bc + 3ca\]

\[ = 3\left( ab + bc + ca \right)\]

\[ = \text { RHS }\]

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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 20 Geometric Progression
Exercise 20.5 | Q 12 | Page 46
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