# How to show an equation with n'th number of iterations

1. Apr 2, 2015

### vmr101

1. The problem statement, all variables and given/known data
An observer measures an object moving away at speed v=0,75, an observer on that object measures another object moving away in the same direction with the same speed relative to it and this is repeated n times. Find the velocity of the n'th object relative to the first.

I didnt post in the physics section, as its just the maths that is the tricky part. Showing this formula for n objects, when each object relies on the same equation for the previous object etc.

2. Relevant equations
w = u+v / (1 + uv) derived from k calculus

3. The attempt at a solution
I have understand and have sorted out the physics in this, but am unsure of how to show the answer mathematically. For large n, g => 1.

g = (a+b) / (1+ab) , where g) is the velocity of the n'th object relative to the original observer.
b) is the velocity of the last (n'th) object relative to the previous, a) is the previous object velocity relative to the one before it, all the way back to the original.
In a few steps I keep subbing in this equation into it self, and while I can show it works for small n, It am unsure of how to show this mathematically for large n.

2. Apr 2, 2015

### vmr101

I recall something like =(a+b)(1-ab+(ab)^2 - (ab)^3+...

3. Apr 2, 2015

### Dick

Look here http://en.wikipedia.org/wiki/Velocity-addition_formula at rapidities. Combining velocities gets complicated. Adding rapidities is easy, they just add. Give an expression using hyperbolic tangents.

4. Apr 4, 2015

### vmr101

I had a look at rapidities but we haven't gone through them so I dont think that's how they want us to show this.
Any other advice?

5. Apr 4, 2015

### Dick

I guess I don't know any other closed form to express the answer in. Rapidities are easy.

6. Apr 5, 2015

### vmr101

I read up on the rapidities and i can make it work :) Thanks Dick.

7. Apr 5, 2015

### Dick

Good for you. I knew you'd like the solution when you figured it out.

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