JEE Advanced2007MathematicsSequences and SeriesActual
Paragraph: Let A₁, G₁, H₁ denote the arithmetic, geometric and harmonic means respectively, of two distinct positive numbers. For n 2 , let A_ n-1 and H_ n-1 has arithmetic, geometric and harmonic means as A_n, G_n, H_n , respectively. Question: Which one of the following statements is correct?
Options
- AG₁>G₂>G₃>
- BG₁ < G₂ < G₃ <
- CG₁=G₂=G₃=
- DG₁ G₄>G₆>
Correct answer
C. G₁=G₂=G₃=
Step-by-step solution
A₁= a+b 2 , G₁= a b and H₁= 2 a b a+b aligned A_n= A_ n-1 +H_ n-1 2 , G_n & = A_ n-1 H_ n-1 , H_n & = 2 A_ n-1 H_ n-1 A_ n-1 +H_ n-1 aligned Clearly, G₁=G₂=G₃= = a b .