1.

(iii) If `a, b c` are respectively the `pth, qth` and `rth` terms of the given `G.P.` then show that `(q-r) log a + (r-p) log b + (p-q)log c = 0`, where `a, b, c > 0. `

Answer» Let A be the first term and R be the common ratio of the given GP. Then,
`a=AR^((p-1)) rArr log a = log A+(p-1) log R` ...(i)
`b=AR^((q-1)) rArr log b = log A+(q-1) log R` ...(ii)
`c=AR^((r-1)) rArr log c= log A+(r-1) log R`. ...(iii)
`:. (q-r) log a+(r-p) log b+(p-q) log c`
`=(q-r) log [AR^((p-1))]+(r-p) log [AR^((q-1))]+(p-q) log [AR^((r-1))]`
`=(log A) {(q-r)+(r-p)+(p-q)}+(log R){(p-1)(q-r)+(q-1) (r-p)+(r-1)(p-q)}`
`=(log A)xx0+(log R)xx0=0`.


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