Question

For each natural number $k$. Let $C_k$ denotes the circle with radius $k$ centimetres and centre at origin. On the circle $C_k$ a particle moves $k$ centimetres in the counter-clockwise direction. After completing its motion on $C_k$ the particle moves to $C_{k+1}$ in the radial direction. The motion of the particle continue in this manner. The particle starts at $(1,0)$. If the particle crosses the positive direction of the $X$-axis for the first time on the circle $C_n$, then $n=\dots$.

Answer 1

It is given that, $C_1$ has centre $(0,0)$ and radius $1.$
Similarly, $C_2$ has centre $(0,0)$ and radius $2$ and $Ck$ has centre $(0,0)$ and radius $k$.
Now, particle starts it motion from $(1,0)$ and moves $1$ radian on first circle then particle shifts from $C_1$ to $C_2$.
After that, particle moves $1$ radian on $C_2$ and then particle shifts from $C_2$ to $C_3$. Similarly, particle move on $n$ circles.
Now, $n\ge 2\pi$ because particle crosses the $X$-axis for the
first time on $C_n$, then $n$ is least positive integer.
Therefore, $n=7$.