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Question Number 7911 by ashis786 last updated on 24/Sep/16

$$\mathrm{If}a>0,b>0,c>0\mathrm{are}\mathrm{respectively}\mathrm{the}$$$$p^{\mathrm{th}},q^{\mathrm{th}},r^{\mathrm{th}}\mathrm{terms}\mathrm{of}\mathrm{a}\mathrm{GP},\mathrm{then}\mathrm{the}\mathrm{value}$$$$\mathrm{of}\mathrm{the}\mathrm{determinant}\left|\begin{array}{ccc}\log a& p& 1\\ \log b& q& 1\\ \log c& r& 1\end{array}\right|\mathrm{is}$$

Answered by prakash jain last updated on 24/Sep/16

$$\mathrm{Let}\mathrm{first}\mathrm{term}\mathrm{of}\mathrm{GP}\mathrm{be}xy\mathrm{with}\mathrm{common}\mathrm{ratio}y$$$$a={xy}^p\Rightarrow \log a=\log x+p\log y$$$$b={xy}^q\Rightarrow \log b=\log x+q\log y$$$$c={xy}^r\Rightarrow \log c=\log x+r\log y$$$$\left|\begin{array}{ccc}\log a& p& 1\\ \log b& q& 1\\ \log c& r& 1\end{array}\right|=\left|\begin{array}{ccc}\log x+p\log y& p& 1\\ \log x+q\log y& q& 1\\ \log x+r\log y& r& 1\end{array}\right|$$$$=\left|\begin{array}{ccc}\log x& p& 1\\ \log x& q& 1\\ \log x& r& 1\end{array}\right|+\left|\begin{array}{ccc}p\log y& p& 1\\ q\log y& q& 1\\ r\log y& r& 1\end{array}\right|$$$$=\log x\left|\begin{array}{ccc}1& p& 1\\ 1& q& 1\\ 1& r& 1\end{array}\right|+\log y\left|\begin{array}{ccc}p& p& 1\\ q& q& 1\\ r& r& 1\end{array}\right|$$$$\mathrm{C}1=\mathrm{C}3\left(\mathrm{in}{1}^{\mathrm{st}}\mathrm{determinant}\right)\mathrm{and}\mathrm{C}1=\mathrm{C}2\mathrm{in}{2}^{\mathrm{nd}}$$$$=\left(\log x\right)\times 0+\log y\times 0=0$$

Commented by Rasheed Soomro last updated on 24/Sep/16

$$Alsolikeyourwayofcorrection.$$

Commented by ashis786 last updated on 24/Sep/16

$$thankyousomuch...$$

Commented by sandy_suhendra last updated on 24/Sep/16

$$ifthefirsttermofGP=xandtheratio=y$$$$soa=the{p}^{th}termofGP={xy}^{p-1}$$$${isn}^{\prime }tit?$$

Commented by prakash jain last updated on 24/Sep/16

$$\mathrm{Thanks}.\mathrm{Updated}\mathrm{the}\mathrm{answer}\mathrm{to}\mathrm{start}\mathrm{with}$$$$\mathrm{first}\mathrm{term}xy.$$

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