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NCERT Class XI Mathematics - Sequences and Series - Solutions

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Question : 56 of 106
Marks: +1, -0
If the pth, qth and rth terms of a G.P. are a, b and c, respectively. Prove that aq–rbr–pcp–qa^{q–r} b^{r–p} c^{p–q} = 1.
Solution:  
Let A be the first term and R be the common ratio, then according to question
ARp–1AR^{p – 1} = a, ARq–1AR^{q – 1} = b, ARr–1AR^{r – 1} = c
∴ aq–rbr–pcp–qa^{q – r} b^{r – p} c^{p – q} = (ARp–1)(q–r)(ARq–1)(r–p)(ARr–1)(p–q)(AR^{p – 1})^(q – r) (AR^{q – 1})^(r – p) (AR^{r – 1})^(p – q)
= A(q–r)+(r–p)+(p–q)A^{(q – r) + (r – p) + (p – q)} R(p–1)(q–r)+(q–1)(r–p)+(r–1)(p–q)R^{(p – 1) (q – r) + (q – 1) (r – p) + (r – 1) (p – q)} = A0R0A^0 R^0 = 1 ⋅ 1 = 1
Hence, aq–rbr–pcp–qa^{q – r} b^{r – p} c^{p – q} = 1.
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