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NCERT Class XI Mathematics - Sequences and Series - Solutions

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Question : 91 of 106
Marks: +1, -0
If a+bxa−bx\frac{a+bx}{a-bx} = b+cxb−cx\frac{b+cx}{b-cx} = c+dxc−dx\frac{c+dx}{c-dx} (x ≠ 0), then show that a, b, c and d are in G.P.
Solution:  
We are given, a+bxa−bx\frac{a+bx}{a-bx} = b+cxb−cx\frac{b+cx}{b-cx} = c+dxc−dx\frac{c+dx}{c-dx} (x ≠ 0)
Applying componendo and dividendo, we get
a+bx+a−bxa+bx−a+bx\frac{a+bx+a-bx}{a+bx-a+bx} = b+cx+b−cxb+cx−b+cx\frac{b+cx+b-cx}{b+cx-b+cx} = c+dx+c−dxc+dx−c+dx\frac{c+dx+c-dx}{c+dx-c+dx}
⇒ 2a2bx\frac{2a}{2bx} = 2b2cx\frac{2b}{2cx} = 2c2dx\frac{2c}{2dx} ⇒ ab\frac{a}{b} = bc\frac{b}{c} = cd\frac{c}{d}
Taking reciprocals, we get ba\frac{b}{a} = cb\frac{c}{b} = dc\frac{d}{c}
Hence, a, b, c, d are in G.P.
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