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

[anemone]

Here is this week's POTW:

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

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[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

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