If a,b,c, all positive ,are pth,qth and rth terms of G.P. , prove that determinant [ log a  p  1 

  log b  q  1  = 0

  log c  r  1 ]

It is given that in a G. P., p^th term = a, q^th term = b and r^th term = c.

Let A and R be the first term and the common ratio of the G.P.

Then,

a = AR ^{p – 1} ⇒ log a = log (AR^{p – 1}) = log A + log R^{p – 1} = log A + (p – 1) log R

i.e., log a = log A + (p – 1) log R

Similarly,

b = AR^{q – 1} ⇒ log b = log A + (q – 1) log R

c = AR^{r – 1} ⇒ log c = log A + (r – 1) log R

               C₂ → C₂ – C₃

     C₁ → C₁ – (log A) C₃ – (log R) C₂

⇒ ∆ = 0

Hence proved.