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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  $a ^{ q-r } b ^{r- p } C ^{p-q} = 1$

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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.

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