Originally Posted by
Soroban Hello, magentarita!
You were quite close . . .
. . . $\displaystyle \begin{array}{cc}\text{principal} & P \,=\,1000 \\$$\displaystyle
\text{amount}& A \,=\,2000 \\
\text{quarterly rate}& \frac{r}{4}\text{ percent} \\
\text{no. of quarters} & n \:=\:32\end{array}$
We have: .$\displaystyle 1000\left(1 + \frac{r}{4}\right)^{32} \:=\:2000 \quad\Rightarrow\quad \left(1 + \frac{r}{4}\right)^{32} \:=\:2$
Take the $\displaystyle 32^{nd}\text{ root: }\;1 + \frac{r}{4} \:=\:2^{\frac{1}{32}} \quad\Rightarrow\quad \frac{r}{4} \:=\:2^{\frac{1}{32}} - 1$
Therefore: .$\displaystyle r \;=\;4\left[2^{\frac{1}{32}}-1\right] \;=\;0.087588595... \;\approx\;8.76\%$