# Find the value of n for circle $C_n$

The problem statement is given like,

For each natural number $k$, let $C_k$ denote the circle with radius $k$ centimetres & centre at the 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 on $C_{k+1}$ in the radial direction. The motion of the particle continues 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 find the value of $n$.

My approach is,

For $C_1$, the particle moves 1 unit. On $C_2$, the particle moves 2 units. Let it crosses the X axis in $n^{th}$ move.

So, in the $n^{th}$ move, it moves n units. Till now it has moved $1+2+3+...(n-1)$ units.

So, when it moves $1+2+3+...(n-1) + n$ units it completes the circumference of the $n^{th}$ circle = $2*\pi*n$

$1+2+3+...(n-1) + n = 2*\pi*n$

$\frac {n*(n+1)}{2} = 2* \pi *n$

$n+1 = 4*\pi$

$n = 4*\pi-1$ = 11.56

Therefore it crosses the x axis for radius = 12.

This approach is definitely wrong because it is not matching with the answers. But what's wrong in my thinking I can't figure it out.

Any kind of help is appreciated. Thanks in advance.

• Moving $k$ centimeter on a circle of radius $k$ is moving by what angle? -- As a sidenote: You did not specify where on $C_1$ the particle starts ... Jul 11, 2017 at 6:43
• @HagenvonEitzen, If I'm not interested in angle and only taking into account the length of arc travelled? Also I'm starting from X axis, that's why I I have considered 1 + 2 + 3 ... + n, to sum up the length of arcs travelled to complete the circumference of the $n_{th}$ circle. That is on the $n_{th}$ circle, it moves gradually distances of 1,2, .. n units till it completes 1 rotation (it's circumference). Jul 11, 2017 at 6:49
• @HagenvonEitzen, I got it.. I am so stupid.. -_-. I'm not considering the "RADIAL JUMP", i.e in my logic, it's jumping parallel to the axis, but actually it jumps in an angle. I missed that angle. Thanks for this. :) Jul 11, 2017 at 6:53

By moving k centimeter on a circle of radius k. We are moving and angle 1 radian (that s the definition on a radian). $$6<2\pi<7$$ Therefore when you cross the positive x-axis you will be on the $C_7$.
To solve this you shoud rather see, that moving $k$ centimeters on the circle with radius $k$cm always makes you move by angle $2\pi\cdot\frac{k}{2k\pi} = 1$. How many times you have to move by angle $1$ to obtain the full angle $2\pi$?