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.