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
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.