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

C_{2} → C_{2} – C_{3}

* C*_{1} → C_{1} – (log A) C_{3} – (log R) C_{2}

⇒ ∆ = 0

Hence proved.

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