Loans and Investments (Part 2)
(This page is Part 2. Click here for Part 1.)
Saving for the Future
You are saving for the future. Your initial deposit is $\,\$4100\,.$ Interest is being earned at an annual rate of $\,5\%\,,$ compounded monthly. You will contribute an additional $\,\$120\,$ each month.
(a)
Find the interest earned
in the first month.
(b)
Write a recursive formula
where $\,u_n\,$ gives the amount
saved (principal plus interest) after $\,n\,$ months.
(c)
Then,
find the amount
saved (principal plus interest)
after $\,7\,$ years.
(d)
Find the total amount of money
you contributed (principal only)
during these $\,7\,$ years.
(e)
Find the total interest earned
during these $\,7\,$ years.
Solution
(a) Find the interest earned in the first month
The interest earned in the first month is: $$\cssId{sb18}{(\$4{,}100)(\frac{0.05}{12}) = \$17.08}$$
(b) Write a recursive formula where $\,u_n\,$ gives the amount saved (principal plus interest) after $\,n\,$ months
The recursive formula is: $$ \begin{align} &\cssId{sb21}{u_0 = 4100}\cr &\cssId{sb22}{u_n = (1 + \frac{0.05}{12})u_{n1} + 120\,,\ \ \text{ for } n \ge 1} \end{align} $$
(c) Find the amount saved (principal plus interest) after $\,7\,$ years
Note that $\,7\,$ years is $\,7(12) = 84\,$ months.
From the calculator, or from the form below: $$\cssId{sb26}{u_{84} = 17853.39}$$
Thus, you have saved (principal plus interest) $\,\$17{,}853.39\,$ after $\,7\,$ years.
The form below computes the amount saved (principal plus interest) after $\,n\,$ equal monthly payments:

Your initial deposit is $\,u_0\,$ (in dollars).
That is, $\,u_0\,$ is the amount saved at time zero (the start of your savings program).

Interest is being earned at an annual rate of $\,i\%\,,$ compounded monthly.
For example, if the interest rate is $\,5\%\,,$ then $\,i = 5\,.$ That is, $\,i\,$ does not include the percent sign.

You are contributing an additional $\,\$C\,$ each month.
For example, if you contribute $\,\$120\,,$ then $\,C = 120\,.$ That is, $\,C\,$ does not include the dollar sign.

$\,u_n\,$ is the amount saved (principal plus interest) after $\,n\,$ monthly contributions:
$$ \begin{align} \cssId{sb38}{u(n)}\cr &\cssId{sb39}{= u_n}\cr &\cssId{sb40}{= {\textstyle\bigl(1 + \frac{i/100}{12}\bigr)u_{n1} + C}\ \text{ for }\ n = 1,2,3,\ldots} \end{align} $$
JavaScript Form:
The amount saved
after $\,n\,$ equal monthly payments
$u_0 =\,$ initial deposit, in dollars:
Do not include commas or dollar sign.
Annual interest rate, $\,i\,,$ as a percent:
Do not include the percent sign.
Amount contributed each month, $\,C\,,$ in dollars:
Do not include commas or dollar sign.
Value of $\,n\,$ for which you want $\,u_n\,$:
By clicking the button below, you will compute
the amount saved (principal plus interest)
after
$\,n\,$ monthly contributions.
You can use an asterisk, ‘*’, to denote multiplication.
(d) Find the total amount of money you contributed (principal only) during these $\,7\,$ years
The total amount of money you contributed (principal only) during these $\,7\,$ years is: $$\cssId{sb59}{\$4100 + 7(12)(\$120) = \$14{,}180.00}$$
(e) Find the total interest earned during these $\,7\,$ years
The total interest earned during these $\,7\,$ years is: $$\cssId{sb62}{\$17{,}853.39  \$14{,}180.00 = \$3,673.39}$$
Using WolframAlpha to Work with Recursive Sequences
Find $\,u_n\,$ for $\,n = 3\cdot 52\,,$ if: $$ \begin{align} &\cssId{sb65}{u_0 = 10,000}\cr &\cssId{sb66}{u_n = (1 + \frac{0.075}{52})u_{n1}  20\,,\ \ \text{ for }\ n \ge 1} \end{align} $$
Step 1
Put the information into WolframAlpha:
u(0) = 10000, u(n) = (1 + 0.075/52)*u(n1)  20
Step 2
WolframAlpha verifies your input as:
As part of your answer, you get this recurrence equation solution, which is a nonrecursive description of the sequence:
Step 3
If you click on the recurrence equation solution, then WolframAlpha puts it in the input box for you. This is much safer than typing it in yourself!
Step 4
Change the value of $\,n\,$ to the desired value. If the value of $\,n\,$ involves computations, then (to ensure correct order of operations) put it inside parentheses, as shown here:
Step 5
Scroll down to the decimal approximation:
So, if you borrow $\,\$10,000\,$ at a $\,7.5\%\,$ annual interest rate and pay back $\,\$20\,$ per week, then after $\,3\,$ years you will still owe $\,\$9025.14\,.$