Given:
Initial value = 400
Interest rate = 5% compounded quarterly.
To find:
The function that gives you the amount of money in dollars, J(t) in t years after the initial deposit.
Solution:
The formula for amount is:
[tex]A=P\left(1+\dfrac{r}{n}\right)^{nt}[/tex]
Where, P is principal, r is the rate of interest in decimals, n is the number of times interest compounded in an year and t is the number of years.
The interest rate is 5% compounded quarterly. So, r=0.05 and n=4.
Substituting [tex]P=400,\ r=0.05,\ n=4[/tex] in the above formula, we get
[tex]A=400\left(1+\dfrac{0.05}{4}\right)^{4t}[/tex]
[tex]A=400\left(1+0.0125\right)^{4t}[/tex]
[tex]A=400\left(1.0125\right)^{4t}[/tex]
The required function notation is:
[tex]J(t)=400\left(1.0125\right)^{4t}[/tex]
Therefore, the amount of money in dollars, J(t) in t years after the initial deposit is [tex]J(t)=400\left(1.0125\right)^{4t}[/tex].