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

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).