# Geometric Sequences and Series A

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```					                    George & Anh Tu      Tuesday 12-1 QUAD1048          (Week 4)

Geometric Sequences and Series

A geometric sequence is a list of numbers arranged so that each term is formed by multiplying the
preceding term by a constant. This constant is called the common ratio.

a, ar, ar2, ar3……. arn-1               where: a = first term (a ≠ 0)
r = common ratio

A geometric series is the sum of a geometric sequence.

S = a + ar + ar2 + ar3 + ……. + arn-1

a(r n − 1)                         a(1 − r n )
S=              for r > 1 OR      S=                for r < 1      where: n = number of terms
r−1                                1−r

Example 1
What are the first 5 terms of a geometric sequence where the first term is 4 and the common ratio is
1
? (Ans 4, 2, 1, 0.5, 0.25) What is the sum of the geometric series? (Ans 7.75)
2

Annuity

An annuity can be defined in the following ways:
It consists of payments (or repayments) which are of equal value,
The payments are made at regular time intervals,
For an ordinary annuity, payments are made at the end of each period,
For an annuity due, payments are made at the beginning of each period.

Present value usually applies to the value of loans or mortgages as these are based on the money
borrowed at the start.
Future value usually applies to savings, superannuation and sinking funds.

Ordinary Annuity

Ordinary annuity refers to a stream of equal payments which are made regularly at the end of
each period. We can express the regular stream of payments on the time line below; where the
regular payments, R, are made at the end of each period.

PV                                                                              FV

R            R             R                                                   R

0            1             2           3                                                    n

Using the FV formula, S = R(1+r)n-1 + R(1+r)n-2 + …..+ R(1+r) + R
George & Anh Tu      Tuesday 12-1 QUAD1048      (Week 4)
Rearranging this, S = R + R(1+r) + …..+ R(1+r)n-2 + R(1+r)n-1 . This is the sum of a geometric
series with the first term R and common ratio of (1+r). Applying the formula, we get

 (1 + r ) n − 1 
FV of ordinary annuity = S = R                 
       r        

Where R = regular payment per period, r = interest rate per period, n = number of periods.

Example 2
Suppose you deposit \$200 in a savings account at the end of every year for the next 4 years at 8%
p.a. compounded annually. What is the accumulated amount after 4 years? (Ans \$901.22)

Example 3
When Jonathan entered University, his parents decided that they would reward him upon graduation
with a trip to Europe. They estimated that the trip would cost \$9,000. They wish to make equal
deposits on the last day of each month, starting March 31, 2004, in a bank that pays 12% p.a.
interest compounded monthly. The last deposit will be made on December 31, 2008. How much
should they deposit each month? (Ans \$115.25)

The formula for the present value of an ordinary annuity can also be found from the sum of a
geometric series. It is
 1 − (1 + r ) − n 
PV of ordinary annuity = A = R                    
       r          

Example 4
Suppose you can afford to pay \$200 at the end of each year for the next 4 years at 8% p.a.
compounded annually. What is the largest loan you could obtain? (Ans \$662.43)

Example 5
Annual gym membership fees can be paid by two methods. A member can either pay the whole fee
immediately or pay \$12 per week starting at the end of the fourth week of the year. If the club
charges interest for the whole year on unpaid fees and interest is compounded weekly at 8% p.a.,
what immediate payment would make the two options equivalent? (Ans \$563.36)

Annuity Due

An annuity due is a stream of equal payments which are made regularly at the beginning of each
period.

PV                                                                             FV

R            R              R             R                                   R

0            1              2             3                                   n-1           n
George & Anh Tu      Tuesday 12-1 QUAD1048          (Week 4)

The formula for the future value of an annuity due is

 (1 + r ) n − 1 
FV of annuity due = S = R                  (1+r)
       r        

The formula for the present value of an annuity due is

 1 − (1 + r ) − n 
PV of annuity due = A = R                    (1+r)
       r          

Example 6
A manufacturer wishes to establish a sinking fund that will provide \$500,000 to replace some of its
machinery. It immediately pays into the fund a payment of \$42,000 followed by the same amount at
the beginning of every 6 months. How long will it take to reach its goal if interest is 7% p.a.
compounded semi-annually? (Ans 5 years)

Example 7
Olivia wants to have a big party for her 30th birthday that will be in 15 months. She estimates that
the cost of hiring a cruise boat on Sydney Harbour and catering expenses will be \$3,200. Olivia
begins to save for the party by making monthly deposits, the first made immediately, into a savings
account which has an interest rate of 6% p.a. compounded monthly. How much should each deposit
be? (Ans \$147.30)

Sinking Fund
A sinking fund is a fund into which periodic payments are made in order to satisfy a future
obligation. The future value of an ordinary or annuity due formula is used.

Example 8
A fish-monger wishes to establish a sinking fund to provide \$350,000 to replace his fishing trawler
in 4 years time. He opens a saving account and decides to make equal deposits at the end of every
quarter. How much is each deposit if the rate of interest is 9% p.a. compounded quarterly? (Ans
\$18,415.82)

Example 9 (Continued from example 8)
After making deposits for 3 years, there is a change in the interest rate. If the new rate is 6% p.a.
compounded quarterly, what will the new quarterly deposit be if the fish-monger is to achieve its
\$350,000 target as planned? (Ans \$20,565.78)

Loan Amortisation
A loan is amortised when both the principal and the interest are paid back by a series of equal
periodic payments. That is, each periodic payment can be split into an interest and principal
component.

Example 10
Joseph borrowed \$8,000 at an interest rate of 12% p.a. compounded annually which is to be repaid
over 4 years with equal payments due at the end of each year. Complete the table.
George & Anh Tu         Tuesday 12-1 QUAD1048        (Week 4)
Year end                 (a)                       (b)                    (c)            Principal repaid
Principal outstanding at          Repayment       Interest on principal     (b) – (c)
beginning of period                               = 0.12 x (a)
1
2
3                  4,451.37                 2,633.88                534.26                2,099.72
4                  2,351.65                 2,633.88                282.20                2,351.68
8,000.03

Principal Outstanding

Principal outstanding is the amount of money still owed at a certain point during the loan period. It
is equivalent to the present value of the payments yet to be made.

 1 − (1 + r ) − p 
PO = R                         where p = number of payments yet to be made
       r          

Important Rules

Interest paid in any period = interest rate x principal outstanding at beginning of that period

Principal repaid in any period = payment – interest paid

The total interest paid during a loan is called the finance charge:
Finance charge = total amount paid – loan amount = nR - A

Example 11 (modified from QMA final examination, Nov 2002)
William borrows \$100,000 from a bank to buy a new speed boat and agrees to repay the debt by
quarterly instalments at the end of each quarter over a period of 10 years. If the interest rate is 9%
p.a. compounded quarterly,

a) Compute his quarterly repayment. (Ans \$3,817.74)
b) Compute his principal outstanding at the beginning of the 17th quarter (this means he has just
paid the 16th instalment. (Ans \$70,204.56)
c) Compute the interest component in the 17th instalment. (Ans \$1,579.60)
d) Compute the principal repaid in the 17th instalment. (Ans \$2,238.14)
e) After 5 years, the interest rate increases to 12% p.a. compounded quarterly. If the debt must be
paid off by the original date agreed upon, find the new quarterly instalment. (Ans \$4,096.48)
f) Compute the total amount paid over 10 years. (Ans \$158, 284.40)
g) Compute the finance charge on the loan. (\$58, 284.40)

Example 12 (modified from QMA final examination, Nov 2002)
A pie maker is estimated to yield a net annual return of \$75,000 for the next 10 years at which time
it can be sold for \$10,000. The company wants to earn 9% on its investment and also set up a
sinking fund to replace the purchase price. If money is placed in the fund at the end of each year
and earns 5% p.a. compounded annually, find the price the company should pay for the pie maker.
(Ans \$447,156.34)

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