[tex] \large \mathcal{ANSWER:} [/tex]
[tex] \quad \large \boxed{\$ 820.20} [/tex]
[tex] \large \mathcal{SOLUTION:} [/tex]
[tex] \begin{array}{l} \large \underline{\textsf{Compound Interest Formula}} \\ \\ \large \quad \quad A = P\left(1 + \dfrac{r}{n}\right)^{nt} \\ \textsf{where:} \\ \begin{aligned} \quad A&=\textsf{Future Value} \\ P&=\textsf{Principal(Initial Value)} \\ r&=\textsf{Interest Rate} \\ n&=\textsf{number of times compounded in one}\:“t” \\ t&=\textsf{time} \end{aligned} \end{array} [/tex]
[tex] \begin{array}{l} \large \underline{\textsf{Given:}} \\ \\ \begin{aligned} \quad P &=\$ 200 \:\textsf{(deposit at the start of each quarter)}\\ r&=4 \% = 0.04 \\ n&=4 \\ t_1&= 1\:\textsf{yr (First quarter deposit)} \\ t_2&= \dfrac{3}{4}\:\textsf{yr (Second quarter deposit)} \\ t_3&= \dfrac{1}{2}\:\textsf{yr (Third quarter deposit)} \\ t_4&= \dfrac{1}{4}\:\textsf{yr (Fourth quarter deposit)} \end{aligned} \end{array} [/tex]
[tex] \begin{array}{l} \large \underline{\textsf{Required:}}\\ \\ \quad \textsf{Total Amount} = A_1 + A_2 + A_3 + A_4 \\ \\ \large \underline{\textsf{Solution:}} \\ \\ \begin{aligned} A_1 &= 200\left(1 + \dfrac{0.04}{4}\right)^{4(1)} = \$ 208.12 \\ A_2 &= 200\left(1 + \dfrac{0.04}{4}\right)^{4(\frac{3}{4})} = \$ 206.06 \\ A_3 &= 200\left(1 + \dfrac{0.04}{4}\right)^{4(\frac{1}{2})} = \$ 204.02 \\ A_4 &= 200\left(1 + \dfrac{0.04}{4}\right)^{4(\frac{1}{4})} = \$ 202 \end{aligned} \\ \begin{aligned} \footnotesize \textsf{Total Amount} &= \footnotesize \$ 208.12 + \$ 206.06 + \$ 204.02 + \$ 202 \\ &=\boxed{\$ 820.20} \end{aligned} \end{array} [/tex]
[tex] \texttt \color{cyan} {\#CarryOnLearning} [/tex]
