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Question
If 5th, 8th and 11th terms of a G.P. are p. q and s respectively, prove that q2 = ps.
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Solution
\[\text { Let a be the first term and r be the common ratio of the given G . P } . \]
\[ \therefore p = 5^{th}\text { term } \]
\[ \Rightarrow p = a r^4 . . . \left( 1 \right)\]
\[q = 8^{th} \text { term } \]
\[ \Rightarrow q = a r^7 . . . \left( 2 \right)\]
\[s = {11}^{th} \]
\[ \Rightarrow s = a r^{10} . . . \left( 3 \right)\]
\[\text { Now, } q^2 = \left( a r^7 \right)^2 = a^2 r^{14} \]
\[ \Rightarrow \left( a r^4 \right) \left( a r^{10} \right) = ps \left[ \text { From } \left( 1 \right) \text { and } \left( 3 \right) \right]\]
\[ \therefore q^2 = ps\]
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