If p^th , q^th and r^th terms of a G.P are a, b and c respectively. Prove that a^{q - r} × b^{r - p} × c^{p - q} = 1

Solution:

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

According to the given condition,

AR^{p - 1} = a

AR^{q - 1} = b

AR^{r - 1} = c

Therefore,

a^{q - r} × b^{r - p} × c^{p - q} = A^{q - r} x R^{( p - 1)(q - r)} x A^{r - p} x R^{(q - 1)(r - p)} x A^{p - q} x R^{(r - 1)(p - q)}

= A^{q - r + r - p + p - q} x R^{( pr - pr - q + r) +(rq - r + p - pq) + ( pr - p- qr + q)}

= A⁰ x R⁰

= 1

Hence proved

NCERT Solutions Class 11 Maths Chapter 9 Exercise 9.3 Question 22

If p^th , q^th and rth terms of a G.P are a, b and c respectively. Prove that a^{q - r} × b^{r - p} × c^{p - q} = 1.

Summary:

Given that the p^th , q^th and rth terms of a G.P are a, b and c we have proved that a^{q - r} × b^{r - p} × c^{p - q} = 1