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The fractional part of a real number x is $x –[x]$, where $[x]$ is the greatest integer less than or equal to $x.$ Let $F_1$ and $F_2$ be the fractional parts of $(44-\sqrt 2017)^{2017}$ and $(44+ \sqrt 2017)^{2017}$ respectively. Then $F_1+F_2$ lies between?

a) $0$ and $0.45$ b) $0.45$ and $0.9$ c) $0.9$ and $1.35$ d) $1.35$ and $1.8$

Any trick to answering the question fast is appreciated (like analyzing the options).

Tapi
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1 Answers1

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Note that $$|44-\sqrt{2017}|\lt1$$ so the absolute value (and hence fractional part) of the first term tends to zero when it is raised to a large exponent. Also we have for all $n\in\mathbb{N}$, $$(44-\sqrt{2017})^n+(44+\sqrt{2017})^n\in\mathbb{N}$$ Hence the fractional part due to the second term must tend to one when raised to a large exponent in order for the above fact to be true. This leads to the conclusion that $F_1+F_2\approx 1$ for large $n$ and actually tends towards the value of $1$ when $n\to\infty$. This means that the only possible answer would be c).

Peter Foreman
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    If $(44+\sqrt{2017})^{2017}=a_1+b_1\sqrt{2017}=n_1+F_1$, then $(44-\sqrt{2017})^{2017}=a_1-b_1\sqrt{2017}=n_2+F_2$. So $F_1+F_2=2a_1-n_1-n_2$ is an integer. As clearly $0<F_i<1$, it follows that $F_1+F_2=1$ exactly. – Hagen von Eitzen Aug 01 '19 at 17:46
  • Yes, sorry, what you have written is correct. I don't think I should edit my answer because it would be to effectively change it to what you have written here. – Peter Foreman Aug 01 '19 at 17:54