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if a1,a2,a3,......an real and greater than are euals to 1.PT (1/1+a1)+(1/1+a2)+......+(1/1+an)is greater than are equals to 1/1+(nth root of a1a2a3....an).

if a1,a2,a3,......an real and greater than are euals to 1.PT (1/1+a1)+(1/1+a2)+......+(1/1+an)is greater than are equals to 1/1+(nth root of a1a2a3....an).

Grade:10

1 Answers

Askiitians Expert Soumyajit IIT-Kharagpur
28 Points
10 years ago

Dear,

ans:- We know that A.M>G.M>HM

or AM>HM

Using the above relation we get

1/n∑(1/1+a1)>n/∑(1+a1)

or ∑(1/1+a1)>n²/(n+∑a1)

Let ∑(1/1+a1)=S

then we Have S>n²/(n+∑a1)....................(1)

Now using AM>GM we get

(1/n)∑a1>(∏a1)^(1/n)...........................(2)

From 1 & 2 we get

S>n²/(n+n(∏a1)^1/n)

or S>n/(1+(n th root of  a1a2a3......an)  (Proved)

 

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SOUMYAJIT IIT_KHARAGPUR

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