Question

If $a,b,c$ are in H.P., then for all $n\in N$ the true statement is

• A. $a^n+c^n<2b^n$
• B. $a^n+c^n>2b^n$
• C. $a^n+c^n=2b^n$
• D. $0$

Answer 1

The correct option is B

$a^n+c^n>2b^n$

Explanation for the correct option:

Given $a,b,c$are in H.P

$$\begin{gathered} H.M,\mathbf{}\mathit{b}\mathbf{=}\frac{\mathbf{2}\mathbf{a}\mathbf{c}}{\mathbf{(}\mathbf{a}\mathbf{+}\mathbf{b}\mathbf{)}} \\ G.M,\mathbf{}\mathit{b}\mathbf{=}\sqrt{\mathbf{a}\mathbf{c}}\dots \left(i\right) \\ AMof{\mathit{a}}^{\mathbf{n}}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}{\mathit{c}}^{\mathbf{n}}\mathbf{=}\frac{\mathbf{(}{\mathbf{a}}^{\mathbf{n}}\mathbf{+}{\mathbf{c}}^{\mathbf{n}}\mathbf{)}}{\mathbf{2}} \end{gathered}$$

We know that $A.M\ge G.M$

$\Rightarrow$ $\frac{(a^n+c^n)}{2}\ge \sqrt{a^n{c}^n}$

$\Rightarrow$ $\frac{(a^n+c^n)}{2}\ge {\left(ac\right)}^{\frac{n}{2}}$

$\Rightarrow$ $\frac{(a^n+c^n)}{2}\ge b^n$

So $(a^n+c^n)\ge 2b^n$

Hence, Option $\mathbf{\left(}\mathit{B}\mathbf{\right)}$ is correct .