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  • If  is integral part of <img alt="\small (2+\sqrt{3})^n" src="https://entrancecorner.oncodecogs.com/gif

If \small I is integral part of \small (2+\sqrt{3})^nand f is fraction part of \small (2+\sqrt{3})^n, then find \small (I+f)(1-f)

Option: 1

0


Option: 2

-1


Option: 3

2


Option: 4

1


Answers (1)

(2+\sqrt{3})^n=1+f where I is an integer and 0 \leq \mathrm{f}<1

Here note that (2-\sqrt{3})^n(2+\sqrt{3})^n=(4-3)^n=1

Since (2+\sqrt{3})^n(2-\sqrt{3})^n=1  it is thus required to prove that  (2-\sqrt{3})^n=1-f 

but,  \begin{aligned} (2-\sqrt{3})^n+(2+\sqrt{ } 3)^n= & {\left[2^n-C_1 \cdot 2^{n-1} \cdot \sqrt{3}+C_2 2^{n-2} \cdot(\sqrt{ } 3)^2-\ldots\right] } \\ & +\left[2^n+C_1 \cdot 2^{n-1} \cdot \sqrt{3}+C_2 2^{n-2} \cdot(\sqrt{ } 3)^2-\ldots\right] \end{aligned}
=2\left[2^n+C_2 \cdot 2^{n-2} \cdot 3+C_4 2^{n-4} \cdot 3^2+\ldots\right]=\text { even integer }

\begin{aligned} & \text { Now } 0<(2-\sqrt{3})<1 \\ & 0<(2-\sqrt{3})^n<1 \\ & \text { if }(2-\sqrt{3})^n=f^{\prime} \text {, then } I+f+f^{\prime}=\text { Even } \\ & \text { Now } O \leq f<1 \text { and } 0<f^{\prime}<1 \\ & \text { Also } I+f+f^{\prime}=\text { Even integer } \\ & f+f^{\prime}=\text { integer } \\ & \text { (1) and }(2) \text { imply that } f+f^{\prime}=1\left(\text { since } 0<f+f^{\prime}<2\right) \\ & \Rightarrow \text { I is odd and } f^{\prime}=1-f \Rightarrow(I+f)(1-f)=1 . \end{aligned}

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Kshitij

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