UNIT 1 :SEQUENCES AND SERIES
Key unit competence: Apply arithmetic and geometric sequences to solve
problems in financial mathematics.
1.0 INTRODUCTORY ACTIVITY
Suppose that an insect population is growing in such a way that each new
generation is 2 times as large as the previous generation. If there are 126
insects in the first generation, on a piece of paper, write down the number
of insects that will be there in second, third, fourth,…nthgeneration.
How can we name the list of the number of insects for different
generations?
1.1 Generalities on sequences
ACTIVITY 1.1
Fold once an A4 paper, what is the fraction that represents the part
you are seeing?
Fold it twice, what is the fraction that represents the part you are
seeing?
What is the fraction that represents the part you are seeing if you
fold it ten times?
What is the fraction that represents the part you are seeing if youfold it n times?
Write a list of the fractions obtained starting from the first until thenth fraction.
The empty sequence { } is included in most notions of sequences, but may
be excluded depending on the context. Usually a numerical sequence is given
by some formulanu = f
, permitting to find any term of the sequence by its
number n; this formula is called a general term formula.
A second way of defining a sequence is to assign a value to the first (or the first
few) term(s) and specify the nth term by a formula or equation that involves
one or more of the terms preceding it. Sequences defined this way are said tobe defined recursively, and the rule or formula is called a recursive formula.
Infinite and finite sequences
Consider the sequence of odd numbers less than 11: This is 1, 3,5,7,9. This is
a finite sequence as the list is limited and countable. However, the sequence
made by all odd numbers is:
1,3,5,7,9,...2n +1,...This suggests the definition that an infinite sequence is a
sequence whose terms are infinite and its domain is the set of positive integers.
Note that it is not always possible to give the numerical sequence by a generalterm formula; sometimes a sequence is given by description of its terms.
In each term, the numerator is the same as the term number, and the denominator
is one greater the term number.
Common difference
The fixed numbers that bind each sequence together are called the common
differences. Sometimes mathematicians use the letter d when referring tothese types of sequences.
1.5.Arithmetic Means of an arithmetic sequence
ACTIVITY 1.5
Suppose that you need to form an arithmetic sequence of 7 terms
such that the first term is 2 and the seventh term is 20. Write down
that sequence given that those terms are 2, A, B,C,D, E, 20 .
If three or more than three numbers form an arithmetic sequence, then all termslying between the first and the last numbers are called arithmetic means. If B
APPLICATION ACTIVITY 1.6
1)Consider the arithmetic sequence 8, 12, 16, 20, … Find the
expression for Sn
2) Sum the first twenty terms of the sequence 5, 9, 13,…
3) The sum of the terms in the sequence 1, 8, 15, … is 396. How many
terms does the sequence contain?
4) Practical activity: A ceramic tile floor is designed in the shape
of a trapezium 10m wide at the base and 5m wide at the top asillustrated on the figure bellow:
The tiles, 10cm by 10cm, are to be placed so that each successive
row contains one less tile than the preceding row. How many tileswill be required?
1.7 Harmonic sequences and its general term
ACTIVITY 1.7
Consider the following arithmetic sequence:
2, 4, 6, 8, 10, 12, 14, 16, …2n,...
a) Form another sequence whose terms are the reciprocals of the
terms of the given sequence.
b) What can you say about the new sequence? What is its first term,
the third term and the general term? Is there a relationship between
two consecutive terms?
Harmonic sequence is a sequence of numbers in which the reciprocals of theterms are in arithmetic sequence. It is of the following form:
Remark
To find the term of harmonic sequence, convert the sequence into arithmetic
sequence then do the calculations using the arithmetic formulae. Then take
the reciprocal of the answer in arithmetic sequence to get the correct term inharmonic sequence.
1. 8 Generalities on Geometric sequence and its general term
ACTIVITY 1.8
Take a piece of paper with a square shape.
1. Cut it into two equal parts.
2. Write down a fraction corresponding to one part according to
the original piece of paper.
3. Take one part obtained in step 2) and repeat step 1) and then step 2)
4. Continue until you remain with a small piece of paper that you
are not able to cut into two equal parts and write down the
sequence of fractions obtained.
5. Observe the sequence of numbers you obtained and give therelationship between any two consecutive numbers.
Sequences of numbers that follow a pattern of multiplying a fixed number rfrom one term u1 to the next are called geometric sequences.
The following sequences are geometric sequences:
1.9.Geometric Means
ACTIVITY 1.9
Suppose that you need to form a geometric sequence of 6 terms such
that the first term is 1 and the sixth term is 243. Given that theseterms are 1, A,B,C,D, 243 . Write down that sequence.
1. 10. Geometric series
ACTIVITY 1.10
During a competition of student teachers at the district level, 5 first
winners were paid an amount of money in the way that the first got
100,000Frw, the second earned the half of this money, the third got
the half of the second’s money, and so on until the fifth who got the
half of the fourth’s money.
a) Discuss and calculate the money earned by each student from the
second to the fifth.
b) Determine the total amount of money for all the 5 student teachers.
c) Compare the money for the first and the fifth student and discuss
the importance of winning at the best place.
d) Try to generalize the situation and guess the money for the
student who passed at the nth place if more students were paid. Inthis case, evaluate the total amount of money for n students.
1.12 Application of sequences in real life
ACTIVITY 1.12
Carry out a research in the library or on internet and find out at least
3 problems or scenarios of the real life where sequences and series are applied.
There are many applications of sequences. Sequences are useful in our daily lives
as well as in higher mathematics. For example; the monthly payments made to
pay off an automobile or home loan with interest portion, the list of maximum
daily temperatures in one area for a month are sequences. Sequences are used
in calculating interest, population growth, half-life and decay in radioactivity,
etc.
In economics and Finance, sequences and series can be used for example in
solving problems related to:
a) Final sum, the initial sum, the time period and the interest rate for an
investment.
The amount A after t years due to a principal P invested at an annual interestrate r compounded n times per year is
value at t = 0 ; r is the Interest rate expressed as a decimal; r is the number of
years P is invested; A is the amount after t years.
The amount A after t years due to a principal P invested at an annual interestrate r compounded continuously is
7) To save for her daughter’s college education, Martha decides to put $50
aside every month in a bank guaranteed-interest account paying 4% interest
compounded monthly.
She begins this savings program when her daughter is 3 years old. How much
will she have saved by the time she makes the 180th deposit? How old is herdaughter at this time?
APPLICATION ACTIVITY 1.12
1) If Linda deposits $1300 in a bank at 7% interest compounded
annually, how much will be in the bank 17 years later?
2) The population of a city in 1970 was 153,800. Assuming that
the population increases continuously at a rate of 5% per year,
predict the population of the city in the year 2000.
3) To save for retirement, Manasseh, at age 35, decides to place
2000Frw into an Individual Retirement Account (IRA) each year
for the next 30 years. What will the value of the IRA be when
Manasseh makes his 30th deposit? Assume that the rate of return
of the IRA is 4% per annum compounded annually.
4) A private school leader received permission to issue 4,000,000Frw
in bonds to build a new high school. The leader is required to
make payments every 6 months into a sinking fund paying 4%
compounded semiannually. At the end of 12 years the bondobligation will be retired. What should each payment be?
