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If A, B, C Are in G.P., Prove That: 1 a 2 − B 2 + 1 B 2 = 1 B 2 − C 2 - Mathematics

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Question

If a, b, c are in G.P., prove that:

\[\frac{1}{a^2 - b^2} + \frac{1}{b^2} = \frac{1}{b^2 - c^2}\]

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Solution

a, b and c are in G.P.

\[\therefore b^2 = ac\]   .......(1)

\[\text {  LHS } = \frac{1}{a^2 - b^2} + \frac{1}{b^2}\]

\[ = \frac{b^2 + a^2 - b^2}{\left( a^2 - b^2 \right) b^2}\]

\[ = \frac{a^2}{\left( a^2 b^2 - b^4 \right)}\]

\[ = \frac{a^2}{a^2 \left( ac \right) - \left( ac \right)^2}\]

\[ = \frac{1}{ac - c^2}\]

\[ = \frac{1}{b^2 - c^2} = \text { RHS }\]

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

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RD Sharma Mathematics [English] Class 11
Chapter 20 Geometric Progression
Exercise 20.5 | Q 8.4 | Page 46

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