If the pth, qth and rth terms of a G.P. are a, b and c, respectively. Prove that aq-r br-p cp-q = 1

Question

If the p^th, q^th and r^th terms of a G.P. are a, b and c, respectively. Prove that a^q-rb^r-pc^p-q = 1

Asked by Pragya Singh | 1 year ago | 97

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Class 11 Maths Sequences and Series Add Answer

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Accepted Answer

Let’s take A to be the first term and R to be the common ratio of the G.P.

Then according to the question, we have

AR^p–1= a

AR^q–1= b

AR^r–1= c

Then,

a^q–rb^r–pc^p–q

= A^q–r× R^{(p–1) (q–r)} × A^r–p × R^{(q–1) (r–p)} × A^p–q × R^{(r –1)(p–q)}

= Aq^{– r + r – p + p – q} × R ^{(pr – pr – q + r) + (rq – r + p – pq) + (pr – p – qr + q)}

= A⁰ × R⁰

= 1

Hence proved.

Answered by Abhisek | 1 year ago