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If three positive unequal numbers a,b,c are in HP,then a.(a+c)>2b b.a 2 +c 2 >2b 2 c.a 2 +c 2 >2ac d.none of these

If three positive unequal numbers a,b,c are in HP,then


a.(a+c)>2b


b.a2+c2>2b2


c.a2+c2>2ac


d.none of these

Grade:12th Pass

1 Answers

Ashwin Muralidharan IIT Madras
290 Points
9 years ago

Hi Menka,

 

All of them will be true.

As a,b,c are in HP, we have 2/b = 1/a+1/c.

So 4/b2 = 1/a2+1/c2+2/ac -------(1).

 

Now using AM-GM Inequality

a2+c2 > 2ac -----{the equality is removed as a,c cannot be equal as given in the question}

1/a2+1/c2 > 2/ac.

Now from (1) 4/b2 > 2/ac + 2/ac

or 4/b2 > 4/ac, or ac>b2

So a2+c2 > 2b2.

 

Next a+c > 2√(ac) > 2b.

 

Hence all of them are true.

 

Hope it helps.

 

Best Regards,

Ashwin (IIT Madras).

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