Question

If $l,m$ and $n$ are the $p^{th}$, $q^{th}$ and $r^{th}$ terms of a G.P. and are all positive, then
$\left|\begin{array}{ccc}\ln l& p& 1\\ \ln m& q& 1\\ \ln n& r& 1\end{array}\right|$ equals to

Answer 1

$l=A{R}^{p-1}\Rightarrow \ln l=\ln A+(p-1)\ln R\Rightarrow m=A{R}^{q-1}\Rightarrow \ln m=\ln A+(q-1)\ln R\Rightarrow n=A{R}^{r-1}\Rightarrow \ln n=\ln A+(r-1)\ln R$
Now, $\left|\begin{array}{ccc}\ln l& p& 1\\ \ln m& q& 1\\ \ln n& r& 1\end{array}\right|=\left|\begin{array}{ccc}\ln A+(p-1)\ln R& p& 1\\ \ln A+(q-1)\ln R& q& 1\\ \ln A+(r-1)\ln R& r& 1\end{array}\right|=\ln R\left|\begin{array}{ccc}p-1& p& 1\\ q-1& q& 1\\ r-1& r& 1\end{array}\right|\text{Using}{C}_1\to C_1-\ln A\cdot C_3)=\ln R\left|\begin{array}{ccc}p& p& 1\\ q& q& 1\\ r& r& 1\end{array}\right|=0(C_1\to C_1+C_3)$
$\because C_1$ and $C_2$is equal.