Answer:
$95.5090 must be deposited monthly
Step-by-step explanation:
From the information given:
The annual interest rate (r) = 4.2% = 0.042
Let assume that an amount Y is deposited, then after one month, it will increase to:
[tex]Y ( 1+ \dfrac{0.042}{12})[/tex]
The total amount after 4 years will be:
[tex]= Y ( 1+ \dfrac{0.042}{12})^{48}+Y ( 1+ \dfrac{0.042}{12})^{47} +Y ( 1+ \dfrac{0.042}{12})^{46} +...+ Y ( 1+ \dfrac{0.042}{12})[/tex]
[tex]= Y ( 1.0035)^{48}+Y ( 1.0035)^{47} +Y( 1.0035)^{46} +...+ Y( 1.0035)[/tex]
Using the sum of a geometric progression:
[tex]= Y (1.0035) \dfrac{ (1.0035^{48}-1)}{(1.0035-1)}[/tex]
[tex]= Y (1.0035) \dfrac{ (1.0035^{48}-1)}{0.0035}[/tex]
The above amount is then equal to $5000
i.e
[tex]= Y (1.0035) \dfrac{ (1.0035^{48}-1)}{0.0035} = 5000[/tex]
[tex]Y = \dfrac{5000\times 0.0035}{(1.0035)(1.0035^{48}-1)} \\ \\ \mathbf{Y = \$95.5090}[/tex]
