Question

If for a G.P., p^th, q^th and r^th terms are a, b and c respectively; prove that : (q – r) log a + (r – p) log b + (p – q) log c = 0

Answer 1

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

Then,

p^th term = a `=>` AR^{p – 1} = a

q^th term = b `=>` AR^{q – 1} = b

r^th term = c `=>` AR^{r – 1} = c

Now,

a^{q – r} × b^{r – p} × c^{p – q} = (AR^{p – 1})^{q – r} × (AR^{q – 1})^{r – p} × (AR^{r – 1})^{p – q}

= `A^(q - r) . R^((p - 1)(q - r)) xx A^(r - p) . R^((q - 1)(r - p)) xx A^(p - q) . R^((r - 1)(p - q))`

= `A^(q - r + r - p + p - q) xx R^((p - 1)(q - r) + (q - 1)(r - p) + (r - 1)(p - q))`

= A⁰ × R⁰

= 1

Taking log on both the sides, we get

log (a^{q – r} × b^{r – p} × c^{p – q}) = log 1

`=>` (q – r) log a + (r – p) log b + (p – q) log c = 0    ...(proved)