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Hello to everybody I have a problem because I can't solve this equation:

$$960 - \frac{84.60}{(1+x)^{\frac1{12}}} - \frac{84.60}{(1+x)^{\frac2{12}}} - \cdots - \frac{84.60}{(1+x)^{\frac{11}{12}}} - \frac{84.60}{(1+x)^{\frac{12}{12}}} = 0$$

I haven't any idea about the approach to use in order to obtain "$x$".

Everybody can help me, please?

Thank you

Cameron Buie
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Owl
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    Could you clarify what 84,60 means? And does e.g. 60/(1+x)^1/12 mean $\frac{60}{(1+x)^{1/12}}$? – Rebecca J. Stones Sep 18 '13 at 22:47
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    What you call "equation" is really unclear. Use some LaTeX to write mathematics and clear that up. – DonAntonio Sep 18 '13 at 22:51
  • Sorry but I'm new here. Yes Rebecca it means exactly what you have indicated – Owl Sep 18 '13 at 23:03
  • You still haven't explained what that odd 84 there means, @Owl...and new or not: read a little before posting to learn how to properly do it. – DonAntonio Sep 18 '13 at 23:39
  • I'll read rules as soon as possible. 84.60 are rates of a loan and the unknown x is the TAEG. I have to found the value of x. But I don't know how. Can I use Taylor series in order to simplify the denominator? – Owl Sep 19 '13 at 00:05
  • @Owl: I have attempted to clean up the formatting so that the question is understandable. Please be certain that my edits are true to the intention of your post. – Cameron Buie Sep 19 '13 at 00:24
  • Perfect, thank you Cameron Buie – Owl Sep 19 '13 at 00:28

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I'll write $(1+x)^{1/12}$ as $u$ to save typing, so your equation would be $$\begin{align} 960&=84.6\left(\frac{1}{u}+\frac{1}{u^2}+\cdots+\frac{1}{u^{12}}\right)\\ &=84.6\left(1+\frac{1}{u}+\frac{1}{u^2}+\cdots+\frac{1}{u^{12}}\right)-84.6 \end{align}$$ The reason I added and subtracted $84.6$ on the right was that the parenthesized expression is now a familiar geometric series with value $$ 1+\frac{1}{u}+\frac{1}{u^2}+\cdots+\frac{1}{u^{12}}=\frac{1-(1/u)^{13}}{1-(1/u)}=\frac{u-(1/u)^{12}}{u-1} $$ so your equation is now (after a trifling bit of algebra) $$ \frac{1044.6}{84.6}=\frac{u-(1/u)^{12}}{u-1} $$ or $$ 960u^{13}-1044.6u^{12}+84.6=0 $$ This has an obvious solution $u=1$ and for the rest you might have to use approximation methods or, what amounts to the same thing, a computer algebra system.

Update. Mathematica tells me there are three real solutions: $u=-0.775465, 1, 1.00871$, corresponding to $x=-0.952712, 0, 0.1096752$, respectively.

Rick Decker
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