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in Geometric Progressions by (15.7k points)

If a, b, c, d are in GP, then prove that

\(\frac{1}{(a^2 + b^2)}, \frac{1}{(b^2 + c^2)}, \frac{1}{(c^2 + d^2)}\) are in GP

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To prove: \(\frac{1}{(a^2 + b^2)}, \frac{1}{(b^2 + c^2)}, \frac{1}{(c^2 + d^2)}\) are in GP.

Given: a, b, c, d are in GP

Proof: When a, b, c, d are in GP then

\(\Rightarrow\) \(\frac{b}{a} = \frac{c}{b} = \frac{d}{c}\)

From the above, we can have the following conclusion 

⇒ bc = ad … (i) 

⇒ b2 = ac … (ii) 

⇒ c2 = bd … (iii)

From eqn. (i) , (ii) and (iii)

From the above equation, we can say that \(\frac{1}{(a^2 + b^2)}, \frac{1}{(b^2 + c^2)}, \frac{1}{(c^2 + d^2)}\) are in GP.

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