PreCalculus II Notes for 12-1 Day 3 : Amortization & Annuity Problems
I. Amortization formula-when you make equal periodic payments on a loan to pay it
off, the loan is said to be amortized.
One of the general formulas you can use to find the amount you still owe is:
r 

An  1 
A P
12  n 1

A0  B
n1
If $B is borrowed at an interest rate of r% (expressed as a decimal) per annum compounded
monthly, the balance A n due after n monthly payments of $P is given by the recursive
sequence above.
The initial loan balance $B. THE BALANCE DUE An after n payments will equal the balance
due previously An1 , plus the interest charged on that amount, reduced by the periodic
payment P.
Example 1: You borrowed $120,000 at 6.5% per annum compounded monthly for 30 years
to purchase a home. The monthly payment is determined to be $758.48.
a. Find a recursive formula for the balance on your home after each monthly payment
r 

has been made. Use the formula
above:
An  1 
An 1  P
n1

12 

b. Use the graphing calculator to view a table showing the balance after each monthly
payment. Determine your balance after the first payment.
Press MODE on your calculator and SEQ,
Press y=
nmin = 0
(note: you are starting at the first term)
u(n)= type your formula here
(note: the u is above the number 7 on your calc(2nd 7)
(also note: you will type (1+.065/12)(u(n-1))-758.48
u(nMin) = type your starting value here120000
c. When will the value be below $100,000?
d. Determine the interest expense when the loan is paid off.
You borrowed $120,000 at 6.5% per annum compounded monthly for 30 years to purchase a
home. The monthly payment is determined to be 758.48.
360 payments
359(758.48)+(______________)(1+.065/12)=___________________-120,000=__________________
e. Suppose you decide to pay an extra $100 each month on the loan, is this worthwhile?
Press MODE on your calculator and SEQ,
Press y=
nmin = 0
(note: you are starting at the first term)
u(n)= type your formula here
(note: the u is above the number 7 on your calc(2nd 7)
(also note: you will type 1.005u(n-1)-858.48
u(nMin) = type your starting value here120000
II. Annuity- is a sequence of equal periodic deposits. The deposits may be annually,
quarterly, monthly, or daily. The amount of an annuity is the sum of all deposits made plus
all interest paid.
Formula:
r 

Ao = M, An = 1   An-1 + P
N

Ao = M = initial amount
An = amount gained after n payments
r = rate
N = number of times per year interest is compounded
An-1 = previous amount
P = periodic rate
The money in the account initially, A0 , is $M; the money in the account after n-1 payments,
r
An1 earns interest
during the nth period; so when the periodic payment of P dollars is
N
added, the amount after n payments, An , is obtained.
Example 2:
On January 1, 1999, you decided to place $45 at the end of each month into an Education IRA.
a. Find the recursive formula that represents the balance at the end of each month if the
rate of return is assumed to be 6% per annum compounded monthly.
r 

An = 1   An-1 + P
N

b. How long will it be before the account exceeds $4000?
Press MODE on your calculator and SEQ,
Press y=
nmin = 0
(note: you are starting at the first term)
u(n)= type your formula here
(note: the u is above the number 7 on your calc(2nd 7)
(also note: you will type 1.005u(n-1)+45
u(nMin) = type your starting value here 0
c. What will the value of the account be in 16 years?