Question

If a,b,c $\in$ H.P then which of the following hold good $(\text{if}n\notin (0,1),a,b,c>0)$

• A. $a^n+c^n>2a^n$
• B. $a^n+c^n<2b^n$
• C. $a^n+c^n>2b^n$
• D. $a^n+c^n>2\times b^n$

Answer 1

The correct option is C $a^n+c^n>2b^n$

As a, b, c are in H.P.
$\therefore$ b is H. M. of a and c
Also A.M. of a and c is $\frac{a+c}{2}$
using A. M. > H. M.
$\frac{a+c}{2}>b$
Again mean of $n^{th}$ power > $n^{th}$ power of their means
$\therefore \frac{a^n+c^n}{2}>{\left(\frac{a+c}{2}\right)}^n>b^n$
$\Rightarrow d^n+c^n>2b^n$