Question

If $a,b,c$ are $p^{th},q^{th},r^{th}$ terms respectively of a G.P. then $\left|\begin{array}{ccc}\mathrm{loga}& p& 1\\ \mathrm{logb}& q& 1\\ \mathrm{logc}& r& 1\end{array}\right|$

Answer 1

$a=p^{th}$ term, let common ratio be k and first term be L

$a=L{k}^{p-1}$

$\Rightarrow loga=logL+(p-1)logk$

$b=L{k}^{q-1}$

$\Rightarrow logb=logL+(q-1)logk$

$c=L(k{)}^{(r-1)}$

$\Rightarrow logL=logL+(r-1)logk$

$\left|\begin{array}{ccc}loga& p& 1\\ logb& q& q\\ logc& r& r\end{array}\right|-loga[q-r]-p[logb-logc]+1[r/logb-qlogc]$

$=[logL+(p-1\left)logk\right](q-r)q-p\left[\right(q-r\left)logk\right]+rlogL+r(q-1)logk-qlogL-q(r-1)logk$

$=logL[q-r-q+r]+logk\left[\right(q-r\left)\right(p-1)-p(q-r)+r(q-1)-q(r-1\left)\right]$

$logc\left(0\right)+logk[r-q+rq-r-qr+r]$

$\left|\begin{array}{ccc}loga& p& 1\\ logb& q& 1\\ logc& r& 1\end{array}\right|=0$.