A Fractional-Part Membership Rule

★ 8/10Quadratic Equations number theory algebra sequences Casework

Problem

A strictly increasing sequence of positive reals $\{x_n\}$ has the property: $x \in \{x_n\}$ iff $x + \{x\}^2$ is an integer, where $\{x\}$ is the fractional part of $x$ . Find the sum of the first $100$ terms.

Answer

Solution

Difficulty	8/10
Topics	Quadratic Equations, number theory, algebra, sequences, Casework

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