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Question
If a, b, c are in A.P. and a, b, d are in G.P., show that a, (a − b), (d − c) are in G.P.
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Solution
\[\text { a, b and c are in A . P } . \]
\[ \therefore 2b = a + c . . . . . . . (i)\]
\[\text { Also, a, b and d are in G . P } . \]
\[ \therefore b^2 = ad . . . . . . . (ii)\]
\[\text { Now }, \left( a - b \right)^2 = a^2 - 2ab + b^2 \]
\[ \Rightarrow \left( a - b \right)^2 = a^2 - a\left( a + c \right) + ad \left[ \text { Using } (i)\text { and } (ii) \right]\]
\[ \Rightarrow \left( a - b \right)^2 = a^2 - a^2 - ac + ad\]
\[ \Rightarrow \left( a - b \right)^2 = ad - ac\]
\[ \Rightarrow \left( a - b \right)^2 = a(d - c)\]
\[\text { Therefore, }a, \left( a - b \right) \text { and } (d - c) \text { are in G . P }. \]
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