If  $a,b,c$  are the  $p^{th},q^{th},r^{th}$ terms of G.P then  $(q-r)lna+(r-p)lnb+(p-q)lnc$ is equal to 

$a=\alpha .{\beta }^{p-1},b=\alpha .{\beta }^{q-1}c=\alpha .{\beta }^{r-1}$

Where $T_1=\alpha$, common ratio  $\left(r\right)=\beta ,$

$$\begin{gathered} a=\alpha .{\beta }^{p-1} \\ \ln a=lna+(p-1)ln\beta \\ \ln b=lna+(q-1)ln\beta \\ \ln c=lna+(r-1)ln\beta \\ subin \\ (q-r)lna+(r-p)lnb+(p-q)lnc \\ =0 \end{gathered}$$