1 SEQUENCES AND SERIES

2 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

3 4.1: SEQUENCES AND SERIES

4 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

5 SEQUENCES A sequence is a set of real numbers a 1, a 2,…a n,… which is arranged (ordered). Example: Each number a k is a term of the sequence. We called a 1 - First term and a 45 - Forty-fifth term The nth term a n is called the general term of the sequence.

6 INFINITE SEQUENCES An infinite sequence is often defined by stating a formula for the nth term, a n by using {a n }. Example: The sequence has nth term. Using the sequence notation, we write this sequence as follows First three terms Fifth teen term

7 EXERCISE 1 : Finding terms of a sequence List the first four terms and tenth term of each sequence: ABCABC DEFDEF

8 RECURSIVELY DEFINED SEQUENCES A sequence is said to be defined recursively if the first term a 1 is state together with a rule for obtaining any term a k +1 from the preceding term a k whenever k ≥ 1. Example: A sequence is defined recursively as follows Thus the sequence is 3, 6, 12, 24, … where

9 EXERCISE 2 : Write down the next three terms of the sequence given by the following: ABCABC

10 PERIODIC SEQUENCES A periodic sequence is a sequence with terms which are repeated after a certain fixed number of term. Example: 1. 2.

11 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

12 EXERCISE 3 : Evaluating a sum Find the following sum: ABCABC

13 SERIES In general, given any infinite sequence, a 1, a 2,…a n,… the expression is called an infinite series or simply a series.

14 EXERCISE 4 : Evaluating a series Find the nth term, the number of terms and express each the following series by using the sigma notation. ABCABC

15 EXERCISE 5 : Express series by sigma notation Write down all the terms for each of the following series and hence, find its sum: ABAB C

16 THEOREM OF SUMS Sum of a constant Sum of 2 infinite sequences

17 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 S n. The sequence S 1, S 2,…S n,… is called a sequence of partial sums. nth partial sum

18 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, …

19 4.2: ARITHMETIC SERIES

20 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

21 ARITHMETIC SEQUENCES A sequence a 1, a 2,…a n,… 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.

22 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, …, n, …

23 THE nth TERM OF AN ARITHMETIC SEQUENCES An arithmetic sequence with first term a 1 and common different d, can be written as follows: The nth term, a n of this sequence is given by the following formula:

24 EXERCISE 8 : Finding the nth terms of an arithmetic sequence Find formulas (the nth terms) for the following arithmetic sequences A. 1, 3, 5, 7, 9, … B. 16, 13, 10, 7,… C. -6, -4.5, -3, -1.5

25 EXERCISE 9: Finding a specific term of an arithmetic sequence A. The first three terms of an arithmetic sequence are 10, 16.5, and 13. Find the fifteenth term. B. 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. C. If the fourth term of an arithmetic sequence is 5 and the ninth term is 20, find the sixth term.

26 THE nth PARTIAL SUM OF AN ARITHMETIC SEQUENCES If a 1, a 2,…a n,… is an arithmetic sequence with common difference d, then the nth partial sum S n (that is the sum of the first nth terms) is given by either or

27 EXERCISE 10 : Finding a sum of Arithmetic Sequence A. Find the sum of the first 50 terms of an arithmetic sequence 2, 4, 6,…2n, …. B. Find the sum of integers which lie between 100 and 500 and is divisible by 7. C. Express the following sequence in terms of summation notation:

28 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 c 1, c 2,…c k are real numbers such that a,c 1, c 2,…c k,b is a finite arithmetic sequence, then c 1, c 2,…c k are k arithmetic means between the numbers a and b.

29 EXERCISE 11 : Inserting Arithmetic Means A. Insert three arithmetic means between 2 and 9 B. Insert three arithmetic means between 3 and -5

30 4.3: GEOMETRIC SERIES

31 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

32 GEOMETRIC SEQUENCES A sequence a 1, a 2,…a n,… is a geometric sequence if a 1 ≠ 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.

33 EXERCISE 12: Showing that a sequence is geometric Show that the following sequences are geometric and find the common ratio. A B

34 THE nth TERM OF AN ARITHMETIC SEQUENCES A geometric sequence with first term a 1 and common ratio r, can be written as follows: The nth term, a n of this sequence is given by the following formula:

35 EXERCISE 13 : Finding the nth terms of an geometric sequence Find formulas (the nth terms) for the following arithmetic sequences A. B. C.

36 EXERCISE 14: Finding a specific term of a geometric sequence A. The geometric sequence has first term 3 and common ratio -1/2. Find the first five terms and tenth term. B. The third term of a geometric sequence is 5, and the sixth term is -40. Find the eighth term. C. 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.

37 THE nth PARTIAL SUM OF AN GEOMETRIC SEQUENCES If a 1, a 2,…a n,… is a geometric sequence with common ratio r ≠ 0, then the nth partial sum S n (that is the sum of the first nth terms) is given by

38 EXERCISE 15 : Finding a sum of Geometric Sequence A. Find the sum of the first 5 terms of a geometric sequence 1, 0.3, 0.09, 0,027, …. B. 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. C. Express the following sequence in terms of summation notation:

39 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 c 1, c 2,…c k are real numbers such that a,c 1, c 2,…c k,b is a finite geometric sequence, then c 1, c 2,…c k are k geometric means between the numbers a and b.

40 EXERCISE 16 : Inserting Geometric Means A. Find the geometric means between 20 and 45. B. Insert three geometric means between 2 and 512. C. Find the geometric means of 3 and 4.

41 THE SUM OF AN INFINITE GEOMETRIC SERIES If |r| < 1, then the infinite geometric series has the sum

42 EXERCISE 17: Find the sum of infinite geometric series A. The following sequence is infinite geometric series. Find the sum B. Find a rational number that corresponds to.

: APPLICATIONS OF ARITHMETIC AND GEOMETRIC SERIES

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

45 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. a 1 = 18 inches a 9 = 24 inches Figure 1

46 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

47 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 Figure 3

48 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

49 4.5: BINOMIAL EXPANSION

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

51 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

52 BINOMIAL THEOREM The binomial theorem states that a binomial expansion can be expanded as follows: Where is called a binomial coefficient with

53 PROPERTIES OF BINOMIAL THEOREM 1.There are n + 1 terms in expansion, first being and the last. 2.The power of a decrease by 1 and the power of b increase by 1 along the expansion. 3.The sum of powers of a and b in each term is always equal to n. 4.The (n + 1)th term is.

54 EXERCISE 18 : Finding a Binomial expansion Expand the following by using Binomial theorem: ABCABC DEFDEF

55 EXERCISE 19 : Finding a specific term of a Binomial expansion Without expanding completely, find the indicated term(s) in the expansion of the expression. ABCABC

PASCAL’S TRIANGLE Used to obtain binomial coefficients

57 EXERCISE 20 : Using Pascal’s Triangle Expand the following by using Pascal Triangle ABCABC

58 THANK YOU