NCERT Class XI Mathematics - Sequences and Series - Solutions

Question : 56

Total: 106

If the pth, qth and rth terms of a G.P. are a, b and c, respectively. Prove that aq–rbr–pcp–q = 1.

Solution:

Let A be the first term and R be the common ratio, then according to question
ARp–1 = a, ARq–1 = b, ARr–1 = c
∴ aq–rbr–pcp–q = (ARp–1)(q–r)(ARq–1)(r–p)(ARr–1)(p–q)
= A(q–r)+(r–p)+(p–q) R(p–1)(q–r)+(q–1)(r–p)+(r–1)(p–q) = A0R0 = 1 ⋅ 1 = 1
Hence, aq–rbr–pcp–q = 1.

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