Question $1(16$ pts$).$

$($a$)$ If a savings bank account pays $\backslash (0\mathrm{.}5\backslash \%\backslash )$ interest every month, what are the nominal and effective interest rates per year?

$($b$)$ How much must you invest now at $\backslash (10\backslash \%\backslash )($annual$)$ interest to accumulate $\backslash (\backslash$$ $500,000\backslash )$ in $20$ years?