Paragraph: Let A 1 , G 1 , H 1 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, geometr

Question

Paragraph: Let A 1 ​ , G 1 ​ , H 1 ​ 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?

TestHub · Accepted Answer

Let f ( x ) = k e x − x ​ ∴ Hence, ​ f ′ ( x ) x f ′′ ( x ) [ f ′′ ( x ) ] x = − l n k ​ f ( − ln k ) ​ = k e x − 1 = 0 = − ln k = k e x = 1 > 0 = 1 + ln k ​ For one root of given equation 1 + ln k He n ce , k ​ = 0 = e 1 ​ . ​ A 2 ​ is A M of A 1 ​ and H 1 ​ and A 1 ​ > H 1 ​ ⇒ A 1 ​ > A 2 ​ > H 1 ​ A 3 ​ is AM of A 2 ​ and H 2 ​ and A 2 ​ > H 2 ​ ⇒ ∴ ​ A 2 ​ > A 3 ​ > H 2 ​ A 1 ​ > A 2 ​ > A 3 ​ > … ​

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