given a total width and a given number of decreasing widths to fit that width, what is the % decrease

Question

Hailing from the programming world here, maths has never been my strongest area.

I have a width (TW), and that width must be divided by a given number(N) of smaller widths which decrease incrementally. They should decrease by the same percentage each time(P). The sum of these widths should be equal to TW.

I need to solve for P.

I have been scribbling down on paper for a few hours now, and I am unable to work out the solution.

Please let me know if the equation is solvable (if not then what is missing), and if so how to solve it?

Thanks guys!

Answer 1

You have to clarify your post. Nevertheless I have a suggestion:

$\sum_{t=0}^N w \cdot (1-P)^t=TW$

w is the smaller width.

Proof the equation according to what you want to calculate. The LHS can be expressed by the formula for the partial sum of a series.

$\sum_{t=0}^N w \cdot (1-P)^t= w\frac{1-(1-P)^{N+1}}{P}=TW$

To solve this equation you have to use a approximation procedure.

Answer 2

Starting from Calculus's answer, assuming that $w$ is known, you have to solve for $P$ $$f(P)=w\frac{1-(1-P)^{N+1}}{P}-TW=0$$ for which, there is no analytical solution (just as Calculus wrote) and you will need some numerical method such as Newton, the problem being that you need a starting value $P_0$ to start iterations.

What you could notice is the value $P$ you look for corresponds to the intersection of the function $(1-P)^{N+1}$ and the straight line $\Big(1-\frac{TW}{w} P\Big)$. I hope that this could help.