Question

If A, B, C are p^th, q^th and r^th terms of a GP respectively then A^{q - r} · B^{r - p} · C^{p - q} = ______.

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Answer 1

If A, B, C are p^th, q^th and r^th terms of a GP respectively then A^{q - r} · B^{r - p} · C^{p - q} = 1.

Explanation:

Let first term and common ratio of a GP are m and n respectively.

Then, A = m(n^{p - 1})     ....(i)

B = m(n^{q - 1})     ...(ii)

and C = m(n^{r - 1})    ...(iii)

Now, A^{q - r} · B^{r - p} · C^{p - q} 

`= "m"^(("q" - "r")) * "n"^(("q" - "r")("p" - 1)) * "m"^(("r" - "p")) * "n"^(("r" - "p")("q" - 1)) * "m"^(("p" - "q")) * "n"^(("p" - "q")("r" - 1))`

`= "m"^(q- r+r- p + p-q) . "n"^(q p - q - rp + r + rq - r - pq + p + pr - p - qr + q)`

= m⁰.n⁰

= 1 × 1 = 1