Question

Regis deposits $5000 into an account for his 10-year-old child. The annual interest rate is 8.5%, compounded monthly. What monthly deposit is need to ensure the account is worth $1 million by the time the child is 55-years-old. Please explain how to get the answer.

Answer 1

P=5000
D= monthly payment
1 st Month -> P+D

2 nd Month -> (P+D)(1+R)+D = P(1+R)+ D((1+R)+1)

3 rd Month --> (P+D)(1+R)+D ) (1+R) = P(1+R)^2 + D((1+R)^2 + (1+R)+1)

4 th Month --> P(1+R)^3 +D((1+R)^3+(1+R)^2 + (1+R)+1)

nth Month ---> P(1+R)^(n-1) +D(1+(1+R)+..........+(1+R)^n-1)

Answer 2

Using geometry sum formula
\[\left(1+(1+R)+\text{...}\text{...}(1+R)^{n-1}\right)=\text{ }\frac{\left(1-(1+R)^n\right)}{1-(1+R)}\]

Answer 3

substitute back

\[P(1+R)^{n-1}+D\left(\frac{1-(1+R)^n}{-R}\right)\]

Answer 4

Plug in number and Solve

P=5000
R=.085/12
n=45*12

set it equal to 1 million and solve for D

Answer 5

\[5000\left(1+\frac{.085}{12}\right)^{539}+D\left(\frac{1-\left(1+\frac{.085}{12}\right)^{540}}{-\frac{.085}{12}}\right)\text{==}1000000\]

I got $124.23

Answer 6

this is how it grows monthly

in excel