Question

a, b, c are in H.P. and n does not lie between 0 and 1 then $a^n+c^n>2b^n$

• A. True
• B. False

Answer 1

Answer: (A)

True
$\frac{a^n+c^n}{2}>{\left(\frac{a+c}{2}\right)}^n$
If n does not lie between 0 and 1
But we know that A.M. > G.M. > H.M.
$\therefore \frac{a+c}{2}$ the A.M. of a and c is >b i.e. H.M. of a and c because a, b, c are given to be in H.P.
$\therefore {\left(\frac{a+c}{2}\right)}^n>b^n$
$\therefore \frac{a^n+c^n}{2}>{\left(\frac{a+c}{2}\right)}^n>b^n$
$\therefore a^n+c^n>2b^n$