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., pth term = a, qth term = b and rth 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 (ARp – 1) = log A + log Rp – 1 = log A + (p – 1) log R

i.e., log a = log A + (p – 1) log R

Similarly,

b = ARq – 1 ⇒ log b = log A + (q – 1) log R

c = ARr – 1 ⇒ log c = log A + (r – 1) log R

               C2 → C2 – C3

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

⇒ ∆ = 0

Hence proved.

 

 

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