Question

14. You deposit $5000 each quarter into an account earning 3.5% interest compounded quarterly.(a) How much will you have in the account in 10 years? $(b) How much total money will you put into the account? $(c) How much total interest will you earn over the 10 years? $

Answer 1

a) The balance of the account follows the next pattern:

`\begin{gathered} f(0)=5000 \\ f(1)=5000(1+0.035)+5000=5000((1+0.035)+1) \\ f(2)=(5000(1+0.035)+5000)(1+0.035)+5000 \\ =5000(1+0.035)^2+5000(1+0.035)+5000 \\ =5000((1+0.035)^2+(1+0.035)+1) \\ \ldots \end{gathered}`

Where the function is f(t), t in quarters of a year.

10 years consists of 40 quarters; then, we need to calculate f(40),

`\begin{gathered} f(40)=5000((1+0.035)^(40)+(1+0.035)^(39)+\cdots+(1+0.035)+1) \\ \Rightarrow f(40)=5000\sum ^(40)_(i\mathop=0)(1+0.035)^i \\ \Rightarrow f(40)\approx5000(88.51) \\ \Rightarrow f(40)\approx442550 \end{gathered}`

After 10 years, we will have approximately $442550

b) During those 10 years, we made 41 deposits of $5000 (taking into account the first deposit at t=0 and the final deposit at t=40)

`41\cdot5000=205000`

The answer is $205000

c)