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I solved it by just trying out various x's, but is there a better way to solve this and how is this 'brute force' way of solving called?

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I am sorry, I don't know the terminology exactly.

brikas
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    Hint: You should attempt to use the notation of a geometric series. – Sean Roberson Oct 29 '22 at 21:02
  • @SeanRoberson I will take a look – brikas Oct 29 '22 at 21:03
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    A caveat on the above hint: while methods of geometric series are indeed the correct approach here, you will ultimately find a polynomial in $\frac{1}{1+x}$ with terms of degree $1$ and $(22+1)$, the solutions of which are unlikely to have closed forms (since that is the case in general for degree $\ge 5$ polynomials). So you will likely have to resort to numerical methods for solving the roots of polynomials. – Jam Oct 29 '22 at 21:15

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This looks like the problem of the interest in fiance calculations.

The equation write $$13500\frac{ (x+1)^{23}-1}{x(x+1)^{22}}=250000$$ Since we cannot find the roots of a polynomial of degree $23$, we need either numerical methods or approximations.

Rewriting it as $$(x+1)^{23}-1-\frac{500}{27}x(x+1)^{22}=0$$ let use the binomial theorem or Taylor series around $x=0$.

Using very few terms , we then need to solve for $x$ $$\frac{121 }{27}x-\frac{4169 }{27}x^2-\frac{22561}{9}x^3=0$$ SO, at the price of a quadratic equation $$x\sim \frac{\sqrt{414373}-379}{12306} =0.0215$$ while the exact solution is $x=0.0208$.

For sure, we could do much better using more terms and using series reversion.

Claude Leibovici
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