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 of the following statements is correct?

Step-by-step solution

Let f(x)=k e^x-x aligned & & f^ (x) & =k e^x-1=0 & & x & =- k & & f^ (x) & =k e^x & Hence, & [f^ (x) ]_ x=- k & =1>0 & & f(- k) & =1+ k aligned For one root of given equation aligned 1+ k & =0 Hence, k & = 1 e . aligned A₂ is A M of A₁ and H₁ and A₁>H₁ A₁>A₂>H₁ A₃ is AM of A ₂ and H ₂ and A ₂> H ₂ array ll & A₂>A₃>H₂ & A₁>A₂>A₃> array