If (a+b), 2b, (b+c) are in HP, then which one of the following is correct?
- A. a, b and c are in AP
- B. a-b, b-c and c-a are in AP
- C. a, b and c are in GP ✓
- D. a-b, b-c and c-a are in GP
Correct Answer: C. a, b and c are in GP
Explanation
If (a+b), 2b, (b+c) are in HP, their reciprocals are in AP. Thus \frac{2}{2b} = \frac{1}{a+b} + \frac{1}{b+c}. Simplifying gives \frac{1}{b} = \frac{a+2b+c}{ab+ac+b^2+bc}. Cross-multiplying yields ab+ac+b^2+bc = ab+2b^2+bc \implies ac = b^2, which signifies that a, b, c are in GP.
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