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if a, b, c are three distinct positive reals in hp and n doesn`t lie between 0 and 1 then prove a^n+c^n>2b^n

if a, b, c are three distinct positive reals in hp and n doesn`t lie between 0 and 1 then prove a^n+c^n>2b^n

Grade:11

1 Answers

Arun
25750 Points
6 years ago
Please find the answer to your question belowWe know that AM>GMAs a,b,c are in HP(an+cn)/2>\sqrt{a^{n}c^{n}}…............(1)But GM>HMso\sqrt{ac}>bor\sqrt{a^{n}c^{n}}>bn..................(2)So from 1 and 2(an+cn)/2>\sqrt{a^{n}c^{n}}>bnan+cn>2bn

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