Question

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

• A. –1
• B. 2
• C. 1

Answer 1

Let A be the first term and R be the common ratio of the G.P. then,
$l=A{R}^{p-1}\Rightarrow logl=logA+(p-1)logR$.......(i)
$m=A{R}^{q-1}\Rightarrow logm=logA+(q-1)logR$ ......(ii)
$n=A{R}^{r-1}\Rightarrow logn=logA+(r-1)logR$ ......(iii)
Multiplying (i), (ii) and (iii) by
(q - r),(r-p) and (p-q) respectively and adding we get, log l(q - r) + log m(r - p) + log n (p - q) = 0
$\therefore \Delta =0$.