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).
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).