Show that for $a \ne 1$, $a > 0$ the sequence $\{x_n\} = n(1-a^{1\over n})$ is increasing

Question

I'm having difficulties with the following problem:

Let: $$ \begin{cases} x_n = n(1-a^{1\over n})\\ a > 0 \\ a \ne 1 \\ n \in \mathbb N \end{cases} $$ Show that $\{x_n\}$ is an increasing sequence.

I've already tried some default ways of proving such things with the help of division/subtraction but couldn't infer the right inequality from that. I've also played around with logarithms without any luck.

Could anyone give me a hint? (If it would be "divide" or "subtract" then please show some initial steps because i've already tried some of those).

Also please find the visualization via this link.

Answer 1

Use, the inequality, $b^m-1>m(b-1)$ where $m>1,b(\ne 1)>0$.

Now, take $b=a^{\frac{1}{n+1}},m=\frac{n+1}{n}$ and we are done using the above inequality.