[tex]~~~~~~~~~~~~ \textit{Amortized Loan Value} \\\\ pymt=P\left[ \cfrac{\frac{r}{n}}{1-\left( 1+ \frac{r}{n}\right)^{-nt}} \right]\implies pymt=P\left[ \cfrac{\frac{r}{n}}{1-\left(\frac{n}{n+r}\right)^{nt}} \right][/tex]
[tex]\begin{cases} P= \begin{array}{llll} \textit{original amount deposited}\\ \end{array}\dotfill & \begin{array}{llll} 857 \end{array}\\ pymt=\textit{periodic payments}\\ r=rate\to 6.5\%\to \frac{6.5}{100}\dotfill &0.065\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{monthly, thus twelve} \end{array}\dotfill &12\\ t=years\dotfill &30 \end{cases}[/tex]
[tex]pymt=857\left[ \cfrac{\frac{0.065}{12}}{1-\left(\frac{12}{12+0.065}\right)^{12 \cdot 30}} \right]\implies pymt=857\left[ \cfrac{\frac{13}{2400}}{1-\left(\frac{2400}{2413}\right)^{360}} \right] \\\\[-0.35em] ~\dotfill\\\\ ~\hfill pymt\approx 5.42~\hfill[/tex]
