If the pth, qth 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.
If the pth, qth and rth terms of a G.P. are a, b and c, respectively. Prove that aq – r br – pcP – q = 1.
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Solution:
Let A be the first term and R be the common ratio of the G.P.
According to the given information,
ARp–1 = a
ARq–1 = b
ARr–1 = c
aq–r br–p cp–q
= Aq–r × R(p–1) (q–r) × Ar–p × R(q–1) (r-p) × Ap–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
Thus, the given equation is proved.
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