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  • If the pth, qth and rth terms of a G.P. are a, b and c, respectively. Prove that a^ q – r b^ r – p c ^ P – q = 1.

22. If the p^{th} , q ^{th} , r ^{th} terms of a G.P. are a, b and c, respectively. Prove that  a ^{ q-r } b ^{r- p } C ^{p-q} = 1

Answers (1)

best_answer

To prove : a ^{ q-r } b ^{r- p } C ^{p-q} = 1

Let A  be the first term and R be common ratio.

According to the given information, we have

a_p=A.R^{p-1}=a

a_q=A.R^{q-1}=b

a_r=A.R^{r-1}=c

L.H.S : a ^{ q-r } b ^{r- p } C ^{p-q}

           =A^{q-r}.R^{(q-r)(p-1)}.A^{r-p}.R^{(r-p)(q-1)}.A^{p-q}.R^{(p-q)(r-1)}

          =A^{q-r+r-p+p-q}.R^{(qp-rp-q+r)+(rq-pq+p-r)+(pr-p-qr+q)}

         =A^0.R^0=1=RHS

Thus, LHS = RHS.

Hence proved.

Posted by

seema garhwal

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