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