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Suppose I want to distribute unknown points $\{\lambda_1,\lambda_2,\dots,\lambda_n\}$ on an interval $[\lambda_\text{min}, \lambda_\text{max}]$ (on the real line) with spacing $\delta\lambda_i:=\lambda_{i+1}-\lambda_i$ given by $$\delta\lambda_i=\lambda_i^2$$ and $\lambda_1=\lambda_\text{min}$ and $\lambda_n=\min\{\lambda_\text{max},\lambda_{n-1}+\lambda_{n-1}^2\}$.

My question is: Can we find $n:=|\{\lambda_i\}|$, the number of such points?

I have tried setting up an implicit equation $$\begin{align} \sum_{i}^{n-1}\delta\lambda_i&=\lambda_\text{max}-\lambda_\text{min}\\ \Rightarrow\quad\sum_{i}^{n-1}\lambda_i^2&=\lambda_\text{max}-\lambda_\text{min} \end{align}$$ but I am not getting anywhere.

FizzleDizzle
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  • Can't you just start with $\lambda_1 = \lambda_{min}$ and then recursively define $\lambda_{i+1} = \lambda_i + \delta\lambda_i$ until you are done? – Klaus Aug 11 '22 at 09:46
  • Yes, I could do that (essentially writing a computer program which uses a loop and returns $n$), but I was looking more for a closed-form expression for $n$ in terms of $\lambda_\text{min}$ and $\lambda_\text{max}$. – FizzleDizzle Aug 11 '22 at 09:58

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