Question

If a, b, c (not equal) are in H.P and n > 1, then $a^n+c^n$ is:

• A. Less than $2b^n$
• B. Less than or equals to $2b^n$
• C. More than $2b^n$
• D. More than or equals to $2b^n$

Answer 1

The correct option is C More than $2b^n$

Let required HP be 4, 6, 12 and n = 2.
Then check for the given options.
Option (c) is correct.

Alternate Solution:
$\frac{a^n+c^n}{2}>{\left(\frac{a+c}{2}\right)}^n$, if n does not lie between 0 and 1.
But we know that A.M. > H.M
b is the H.M. of a and c
$\frac{a+c}{2}>b$
As n > 1 ${\left(\frac{a+c}{2}\right)}^n>b^n$
$\frac{a^n+c^n}{2}>{\left(\frac{a+c}{2}\right)}^n>b^n$
$a^n+c^n>2b^n$. Option (c) is correct.