Question

If $m>1,1^m+3^m+5^m+\dots \dots .+{(2n-1)}^m$ is

• A. $>n^m$
• B. $>n^{m+1}$
• C. $<n^m$
• D. $<n^{m+1}$

Answer 1

The correct option is B

$>n^{m+1}$

Explanation for correct option

$$\begin{gathered} \left[\frac{1^m+3^m+5^m+\dots \dots .+{(2n-1)}^m}{n}\right]>{\left[\frac{1+3+5+7+...+\left(2n-1\right)}{n}\right]}^m \\ \Rightarrow 1^m+3^m+5^m+\dots \dots .+{(2n-1)}^m>\frac{n}{n^m}{\left[1+3+5+7+...+\left(2n-1\right)\right]}^m \end{gathered}$$

$\Rightarrow 1^m+3^m+5^m+\dots \dots .+{(2n-1)}^m>\frac{n}{n^m}{\left[\frac{n}{2}\left(2+2n-2\right)\right]}^m$; [$n$ term is $\frac{n}{2}\left(2a+d\left(n-1\right)\right)$ where $a$ is the first term and $d$ is the common difference]

$$\begin{gathered} \Rightarrow 1^m+3^m+5^m+\dots \dots .+{(2n-1)}^m>n^{1-m}{\left[n^2\right]}^m \\ \Rightarrow 1^m+3^m+5^m+\dots \dots .+{(2n-1)}^m>n^{1-m}{n}^{2m} \\ \Rightarrow 1^m+3^m+5^m+\dots \dots .+{(2n-1)}^m>n^{1+m} \end{gathered}$$

Hence, option B is correct