Annual rate from initial and final capital

Statement

Mrs Avidi invested two million euros in shares. After ten years she owns thirteen million euros. Determine the annual interest rate $r$ she enjoyed.

Solution

With annual compounding $2\,(1+r)^{10}=13$, that is $(1+r)^{10}=\frac{13}{2}=6.5.$ Taking the tenth root: $1+r=6.5^{1/10}\approx 1.2059 \Rightarrow r\approx 0.2059=20.59\%.$

$\boxed{r=6.5^{1/10}-1\approx 20.59\%}$