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If p ,q ,a n dr are inA.P., show that th...

If `p ,q ,a n dr` are inA.P., show that the pth, qth, and rth terms of any G.P. are in G.P.

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Let A be the first term and R the common ratio of a G.P. Then, `a_(p)=AR^(n-1),a_(q)=AR^(q-1),and a_(r)=AR^(r-1)`. We have to prove that `a_(p),a_(q)` and `a_(r)` are in G.P. given that p,q,r and in A.P.
`(a_(q))^(2)=(AR^(q-1))^(2)`
=`A^(2)R^(2q-2)`
`=A^(2)R^(p+r-2) [becausep,q,r` are in `A.P.therefore2q=p+r]`
`=(AR^(p-1))(AR^(r-1))=a_(p)a_(r)`
Hence, `a_(p),a_(q) anda_(r)` are in G.P.
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