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if 1/a,1/b,1/c,1/d are in AP prove that ab + bc +cd = 3ad

if 1/a,1/b,1/c,1/d are in AP prove that 
ab + bc +cd = 3ad

Grade:11

2 Answers

Nishant Vora IIT Patna
askIITians Faculty 2467 Points
7 years ago
mycroft holmes
272 Points
7 years ago
Dividing by abcd, we have to prove that \frac{1}{cd}+\frac{1}{ad}+\frac{1}{ab} = \frac{3}{bc} or
 
that \frac{1}{cd}+\frac{1}{ad}+\frac{1}{ab} + \frac{1}{bc} = \frac{4}{bc}
 
LHS can be factorised as \left(\frac{1}{c}+\frac{1}{a} \right) \left(\frac{1}{b} + \frac{1}{d} \right)
 
Since the given numbers are in AP we have 
\left(\frac{1}{c}+\frac{1}{a} \right) = \frac{2}{b}; \left(\frac{1}{b} + \frac{1}{d} \right) = \frac{2}{c}
 
Hence we get \left(\frac{1}{c}+\frac{1}{a} \right) \left(\frac{1}{b} + \frac{1}{d} \right) = \frac{2}{b} \times \frac{2}{c} = \frac{4}{bc}

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