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Question
If a, b, c are in G.P., then prove that:
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Solution
\[\text{ a, b and c are in G . P }. \]
\[ \therefore b^2 = ac . . . . . . . . (i)\]
\[\text { Now, LHS }= \frac{a^2 + ab + b^2}{bc + ca + ab}\]
\[ = \frac{a^2 + ab + ac}{bc + b^2 + ab} \left[ \text { Using } (i) \right]\]
\[ = \frac{a\left( a + b + c \right)}{b\left( c + b + a \right)}\]
\[ = \frac{a}{b}\]
\[ = \frac{1}{r}\]
\[\text { Here, r = common ratio }\]
\[\text { RHS }= \frac{b + a}{c + b}\]
\[ = \frac{ar + a}{a r^2 + ar}\]
\[ = \frac{a(r + 1)}{ar(r + 1)}\]
\[ = \frac{1}{r}\]
\[ \therefore\text { LHS = RHS }\]
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