High School How Do You Calculate the Floor of 2√xn for the Given Sequence?

[anemone]

Here is this week's POTW:

-----

Find $$\left\lfloor{2\sqrt{x_n}}\right\rfloor$$ given $$x_n=10^{2n}-10^n+1$$ for all $$n\in N$$.

-----

Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[anemone]

Congratulations to the following members for their correct solution:):

1. kaliprasad
2. lfdahl

Solution from kaliprasad:

Spoiler

We have

$x_n = 10^{2n} - 10^n + 1= (10^n - \frac{1}{2})^2 + \frac{3}{4}$

$ (10^n - \frac{1}{2}) \lt \sqrt{x_n} \lt 10^n$

i.e.

$ (2 * 10^n - 1) \lt 2 \sqrt{x_n} \lt 2* 10^n$

hence the given expression

$\lfloor 2 \sqrt{x_n} \rfloor = 2* 10^n-1$