Question

Annabelle wants to make the most economical decision, so she chooses the 3- year car loan so that after the loan is paid off, she’ll be able to invest in a structured savings account. If Annabelle put $200 into a savings account each month with an annual interest rate of 3.2% interest compound monthly, how much money would she have in her account after 2 years?

Answer 1

$\bf ~~~~~~~~~~~~\textit{Future Value of an ordinary annuity}\\ ~~~~~~~~~~~~(\textit{payments at the end of the period}) \\\\ A=pymnt\left[ \cfrac{\left( 1+\frac{r}{n} \right)^{nt}-1}{\frac{r}{n}} \right]$

$\bf \begin{cases} A= \begin{array}{llll} \textit{accumulated amount}\\ \end{array} \begin{array}{llll}\\\end{array}\\ pymnt=\textit{periodic payments}\to &200\\ r=rate\to 3.2\%\to \frac{3.2}{100}\to &0.032\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{monthly, thus twelve} \end{array}\to &12\\ t=years\to &2 \end{cases}$

$\bf A=200\left[ \cfrac{\left( 1+\frac{0.032}{12} \right)^{12\cdot 2}-1}{\frac{0.032}{12}} \right] \\\\\\ A= 200\left( \cfrac{(1.002\overline{66})^{24}-1}{0.002\overline{66}} \right)\implies A\approx 4950.1193$