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xA = yG = zH: Are x, y, z in AP, GP or HP? | JEE

JEE Maths question with a full step-by-step solution.

Question
If A,G,HA, G, H are respectively the A.M., G.M. and H.M. between two positive numbers, and xA=yG=zHxA=yG=zH where x,y,zx, y, z are non-zero positive quantities, then x,y,zx, y, z are in
AA.P.
BG.P.correct
CH.P.
DA.G.P.
Solution
Step 1: For two numbers a,ba,b, A=a+b2A=\dfrac{a+b}{2}, G=abG=\sqrt{ab}, H=2aba+bH=\dfrac{2ab}{a+b}. Then
AH=a+b2⋅2aba+b=ab=G2,AH=\frac{a+b}{2}\cdot\frac{2ab}{a+b}=ab=G^2,
 so A,G,HA,G,H are in G.P. Step 2: From xA=yG=zHxA=yG=zH, the quantities x,y,zx,y,z are proportional to 1A,1G,1H.\dfrac1A,\dfrac1G,\dfrac1H. Step 3: A,G,HA,G,H in G.P. ⇒\Rightarrow their reciprocals in G.P. ⇒x,y,z\Rightarrow x,y,z in G.P. Correct answer: (2)
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