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# 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)}

= A^{0} x R^{0}

= 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

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