Question

If $a,b,c$ be three unequal positive quantities in H.P., then

• A. $a^{100}+c^{100}>2b^{100}$
• B. $a^3+c^3>2b^3$
• C. $a^5+c^5>2b^5$
• D. $a^2+c^2>2b^2$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $a^{100}+c^{100}>2b^{100}$
B $a^3+c^3>2b^3$
C $a^5+c^5>2b^5$
D $a^2+c^2>2b^2$

$b$ is H.M. of $a\&b$
G.M > H.M.
$\Rightarrow \sqrt{ac}>b$

And A.M of $a^n\&c^n$ > G.M. of $a^n\&c^n$
$\frac{a^n+c^n}{2}>\sqrt{a^n{c}^n}\Rightarrow a^n+c^n>2(\sqrt{ac}{)}^n\Rightarrow a^n+c^n>2b^n$