If the pth-qth and rth terms of a G-P- are a- b and c- respectively- Prove that aq-rbr-pcp-q-1-

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Toppr Admin · Accepted Answer

LetfirsttermAandcommondifference-R-tp-atq-btr-c-x2223-x2223-x2223-x2223-x2223-G-P-tp-ARp-x2212-1-atq-ARq-x2212-1-btr-ARr-x2212-1-cthenaq-x2212-r-br-x2212-p-cp-x2212-q-ARp-x2212-1-q-x2212-r-ARq-x2212-1-r-x2212-p-ARr-x2212-1-p-x2212-qA-q-x2212-r-r-x2212-p-p-x2212-a-R-p-x2212-1-q-x2212-1-q-x2212-1-r-x2212-p-r-x2212-1-p-x2212-q-A-x2218-R-x2218-1-x2234-aq-x2212-r-br-x2212-p-cp-x2212-q-1Proved-

If the pth,qth and rth terms of a G.P. are a, b and c, respectively. Prove that aq−rbr−pcp−q=1.

LetfirsttermAandcommondifference(R)tp=atq=btr=c∣∣ ∣ ∣∣G.P.tp=ARp−1=atq=ARq−1=btr=ARr−1=cthenaq−r.br−p.cp−q{ARp−1}q−r{ARq−1}r−p{ARr−1}p−qA(q−r+r−p+p−a).R{(p−1)(q−1)+(q−1)(r−p)+(r−1)(p−q)}A∘.R∘=1∴aq−r.br−p.cp−q=1Proved.

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