SEQUENCES AND SERIES

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SEQUENCES AND SERIES

• CHAPTER 4
• DCT1043

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CONTENT

• 4.1 Sequences and Series
• 4.2 Arithmetic Series
• 4.3 Geometric Series
• 4.4 Application of Arithmetic
• and Geometric Series
• 4.5 Binomial Expansion

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4.1 SEQUENCES AND SERIES
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OBJECTIVE

• At the end of this topic you should be able to
• Define sequences and series
• Understand finite and infinite sequence, finite
  and infinite series
• Use the sum notation to write a series

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SEQUENCES

• A sequence is a set of real numbers a1, a2,an,
  which is arranged (ordered).
• Example
• Each number ak is a term of the sequence.
• We called a1 - First term and a45 - Forty-fifth
  term
• The nth term an is called the general term of the
  sequence.

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INFINITE SEQUENCES

• An infinite sequence is often defined by stating
  a formula for the nth term, an by using an.
• Example
• The sequence has nth term .
• Using the sequence notation, we write this
  sequence as follows

Fifth teen term
First three terms
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EXERCISE 1 Finding terms of a sequence

• List the first four terms and tenth term of each
  sequence

A B C
D E F
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RECURSIVELY DEFINED SEQUENCES

• A sequence is said to be defined recursively if
  the first term a1 is state together with a rule
  for obtaining any term ak 1 from the preceding
  term ak whenever k 1.
• Example
• A sequence is defined recursively as follows
• Thus the sequence is 3, 6, 12, 24, where

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EXERCISE 2

• Write down the next three terms of the sequence
  given by the following

A B C
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PERIODIC SEQUENCES

• A periodic sequence is a sequence with terms
  which are repeated after a certain fixed number
  of term.
• Example

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THE SUMMMATION NOTATION OF SEQUENCES

• The symbol ? (sigma) is called the summation
  sign.
• This symbol will represents the sum of the first
  m terms as follows

The upper limit
The lower limit
Index of summation
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EXERCISE 3 Evaluating a sum

• Find the following sum

A B C
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SERIES

• In general, given any infinite sequence,
  a1, a2,an, the expression
• is called an infinite series or simply a series.

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EXERCISE 4 Evaluating a series

• Find the nth term, the number of terms and
  express each the following series by using the
  sigma notation.

A B C
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EXERCISE 5 Express series by sigma notation

• Write down all the terms for each of the
  following series and hence, find its sum

A B
C
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THEOREM OF SUMS
Sum of a constant
Sum of 2 infinite sequences
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SEQUENCE OF PARTIAL SUMS

• If n is positive integer, then the sum of the
  first n terms of an infinite sequence will be
  denoted by Sn.
• The sequence S1, S2,Sn, is called a sequence of
  partial sums.

nth partial sum
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EXERCISE 6 Finding the term of the sequence of
partial sums

• Find the first four terms and the nth term of the
  sequence of partial sums associated with the
  following sequence of positive integers.

A 1, 2, 3, , n,
B 2, 9, 28,
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4.2 ARITHMETIC SERIES
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OBJECTIVE

• At the end of this topic you should be able to
• Recognize arithmetic sequences and series
• Determine the nth term of an arithmetic sequences
  and series
• Recognize and prove arithmetic mean of an
  arithmetic sequence of three consecutive terms a,
  b and c

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ARITHMETIC SEQUENCES

• A sequence a1, a2,an, is an arithmetic sequence
  if there is a real number d such that for every
  positive integer k,
• The number is
  called the common difference of the sequence.

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EXERCISE 7 Showing that a sequence is arithmetic

• Show that the following sequences are arithmetic
  and find the common difference.

A 1, 4, 7, 10 , 3n - 2,
B 53, 48, 43, , 58 - 5n,
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THE nth TERM OF AN ARITHMETIC SEQUENCES

• An arithmetic sequence with first term a1 and
  common different d, can be written as follows
• The nth term, an of this sequence is given by the
  following formula

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EXERCISE 8 Finding the nth terms of an
arithmetic sequence

• Find formulas (the nth terms) for the following
  arithmetic sequences
• 1, 3, 5, 7, 9,
• 16, 13, 10, 7,
• -6, -4.5, -3, -1.5

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EXERCISE 9 Finding a specific term of an
arithmetic sequence

• The first three terms of an arithmetic sequence
  are 10, 16.5, and 13. Find the fifteenth term.
• The fifth and eleventh terms of an arithmetic
  sequence are 3 and 6 respectively. Find the
  common difference, first term and nth term of
  this arithmetic sequence.
• If the fourth term of an arithmetic sequence is 5
  and the ninth term is 20, find the sixth term.

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THE nth PARTIAL SUM OF AN ARITHMETIC SEQUENCES

• If a1, a2,an, is an arithmetic sequence with
  common difference d, then the nth partial sum Sn
  (that is the sum of the first nth terms) is given
  by either
• or

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EXERCISE 10 Finding a sum of Arithmetic
Sequence

• Find the sum of the first 50 terms of an
  arithmetic sequence 2, 4, 6,2n, .
• Find the sum of integers which lie between 100
  and 500 and is divisible by 7.
• Express the following sequence in terms of
  summation notation

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THE ARITHMETIC MEAN OF AN ARITHMETIC SEQUENCES

• The arithmetic mean of two number a and b
  (average of a and b) is defined by (a b) / 2 .
• Then the following sequence is true if d (b a
  ) / 2 .
• If c1, c2,ck are real numbers such that a,c1,
  c2,ck,b is a finite arithmetic sequence, then
  c1, c2,ck are k arithmetic means between the
  numbers a and b.

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EXERCISE 11 Inserting Arithmetic Means

• Insert three arithmetic means between 2 and 9
• Insert three arithmetic means between 3 and -5

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4.3 GEOMETRIC SERIES
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OBJECTIVE

• At the end of this topic you should be able to
• Recognize geometric sequences and series
• Determine the nth term of a geometric sequences
  and series
• Recognize and prove geometric mean of an
  geometric sequence of three consecutive terms a,
  b and c
• Derive and apply the summation formula for
  infinite geometric series
• Determine the simplest fractional form of a
  repeated decimal number written as infinite
  geometric series

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GEOMETRIC SEQUENCES

• A sequence a1, a2,an, is a geometric sequence
  if a1 ? 0 and if there is a real number r ? 0
  such that for every positive integer k,
• The number is called the common
  ratio of
• the sequence.

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EXERCISE 12 Showing that a sequence is geometric

• Show that the following sequences are geometric
  and find the common ratio.

A
B
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THE nth TERM OF AN ARITHMETIC SEQUENCES

• A geometric sequence with first term a1 and
  common ratio r, can be written as follows
• The nth term, an of this sequence is given by the
  following formula

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EXERCISE 13 Finding the nth terms of an
geometric sequence

• Find formulas (the nth terms) for the following
  arithmetic sequences

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EXERCISE 14 Finding a specific term of a
geometric sequence

• The geometric sequence has first term 3 and
  common ratio -1/2. Find the first five terms and
  tenth term.
• The third term of a geometric sequence is 5, and
  the sixth term is -40. Find the eighth term.
• For a geometric sequence, whose terms are all
  positive, the fifth and seventh terms are 45 and
  5 respectively. Find the common ratio and the
  first term.

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THE nth PARTIAL SUM OF AN GEOMETRIC SEQUENCES

• If a1, a2,an, is a geometric sequence with
  common ratio r ? 0 , then the nth partial sum Sn
  (that is the sum of the first nth terms) is given
  by

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EXERCISE 15 Finding a sum of Geometric Sequence

• Find the sum of the first 5 terms of a geometric
  sequence 1, 0.3, 0.09, 0,027, .
• The second and fifth terms of geometric sequence
  are 24 and 8/9 respectively. Calculate the first
  term, common ratio and the sum of the first 10
  terms.
• Express the following sequence in terms of
  summation notation

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THE GEOMETRIC MEAN OF AN GEOMETRIC SEQUENCES

• The geometric mean of two number a and b (average
  of a and b) is defined by c .
• If the common ratio is r, then
• If c1, c2,ck are real numbers such that a,c1,
  c2,ck,b is a finite geometric sequence, then
  c1, c2,ck are k geometric means between the
  numbers a and b.

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EXERCISE 16 Inserting Geometric Means

• Find the geometric means between 20 and 45.
• Insert three geometric means between 2 and 512.
• Find the geometric means of 3 and 4.

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THE SUM OF AN INFINITE GEOMETRIC SERIES

• If r lt 1 , then the infinite geometric series
• has the sum

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EXERCISE 17 Find the sum of infinite geometric
series

• The following sequence is infinite geometric
  series. Find the sum
• Find a rational number that corresponds to
  .

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4.4 APPLICATIONS OF ARITHMETIC AND GEOMETRIC
SERIES
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OBJECTIVE

• At the end of this topic you should be able to
• Solve problem involving arithmetic series
• Solve problem involving geometric series

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APPLICATION 1 ARITHMETIC SEQUENCE

• A carpenter whishes to construct a ladder with
  nine rungs whose length decrease uniformly from
  24 inches at the base to 18 inches at the top.
  Determine the lengths of the seven intermediate
  rungs.

a1 18 inches
a9 24 inches
Figure 1
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APPLICATION 2 ARITHMETIC SEQUENCE

• The first ten rows of seating in a certain
  section of stadium have 30 seats, 32 seats, 34
  seats, and so on. The eleventh through the
  twentieth rows contain 50 seats. Find the total
  number of seats in the section.

Figure 2
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APPLICATION 1 GEOMETRIC SEQUENCE

• A rubber ball drop from a height of 10 meters.
  Suppose it rebounds one-half the distance after
  each fall, as illustrated by the arrow in Figure
  3. Find the total distance the ball travels.

10
5 5
2.5 2.5
1.25 1.25
Figure 3
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APPLICATION 2 GEOMETRIC SEQUENCE

• If deposits of RM100 is made on the first day of
  each month into an account that pays 6 interest
  per year compounded monthly, determine the amount
  in the account after 18 years.

Figure 4
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4.5 BINOMIAL EXPANSION
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OBJECTIVE

• At the end of this topic you should be able to
• Expand and solve the Binomial series

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BINOMIAL EXPANSION FOR POSITIVE INTEGERS

• Any expression containing two terms is called a
  binomial eg (a b), (x y).
• If n is a positive integer, then a general
  formula for expanding is given by
  the binomial theorem.
• The following special cases can be obtained by
  multiplication

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BINOMIAL THEOREM

• The binomial theorem states that a binomial
  expansion
• can be expanded as follows
• Where is
  called a binomial
• coefficient with

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PROPERTIES OF BINOMIAL THEOREM

• There are n 1 terms in expansion, first being
  and the last .
• The power of a decrease by 1 and the power of b
  increase by 1 along the expansion.
• The sum of powers of a and b in each term is
  always equal to n.
• The (n 1)th term is .

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EXERCISE 18 Finding a Binomial expansion

• Expand the following by using Binomial theorem

A B C
D E F
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EXERCISE 19 Finding a specific term of a
Binomial expansion

• Without expanding completely, find the indicated
  term(s) in the expansion of the expression.

A B C
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PASCALS TRIANGLE

• Used to obtain binomial coefficients

1 1 1 1
2 1 1 3 3
1 1 4 6 4
1 1 5 10
10 5 1 1 6
15 20 15 6 1
. . . . .
. . .
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EXERCISE 20 Using Pascals Triangle

• Expand the following by using Pascal Triangle

A B C
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