Checkout JEE MAINS 2022 Question Paper Analysis : Checkout JEE MAINS 2022 Question Paper Analysis :

If a,b,c are in H.P., then for all n ϵ N the true statement is

(1) an + cn < 2bn

(2) an + cn > 2bn

(3) an + cn = 2bn

(4) None of these

Solution:

Given a,b,c are in HP.

Then HM, b = 2ac/(a+b)

GM, b = √(ac) …(i)

AM of an and cn = (an+cn)/2

We know AM ≥ GM

(an+cn)/2 ≥ √(ancn)

(an+cn)/2 ≥ (ac)n/2

(an+cn)/2 ≥ bn

So (an+cn) ≥ 2bn

Hence option (2) is the answer.

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