Geotrig problem-1

Note that the angular displacement on circle $C_k$ is $\theta_k = \dfrac k{2\pi k}\times 2\pi = 1$ , which is independent of $k$ . If $\Theta_n$ is the sum of all angular displacements from $C_1$ through $C_n$ or $\displaystyle \Theta_n = \sum_{k=1}^n \theta_k = n$ . When the particle crosses the $x$ -axis for the first time, $\Theta_n = n > 2\pi \approx 6.282 \implies n = \boxed 7$ .

The particle moves through an angle of $1$ radian on each circle; it will cross the positive $x$ -axis when the total angle moved reaches $2\pi=6.28\ldots$ radians, ie on the $\boxed{7^{th}}$ circle.