THE CONCEPT OF SEQUENCE AND SERIES

Adaptif Hal : 2 THE PROGRESSIONS Hal.: 2 The Pattern of Sequence and Series Number Basic Competence: Applying the concept of arithmetic sequence and series Indicator : 1.The value of n-th term in an arithmetic sequence is defined by formula 2.The sum of n in term of arithmetic sequence is defined by formula

Adaptif Hal : 3 THE PROGRESSIONS Hal.: 3 When you ride a motor cycle, have you ever look at the speeedometer? In speedometer,there are numbers of 0,20, 40, 60, 80, 100, and 120 which show the speed of your motor cycle. These numbers are un order, starts from the smallest to the biggest with certain pattern, so that it forms a pattern of sequence The Pattern of Sequence and Series Number

Adaptif Hal : 4 THE PROGRESSIONS Hal.: 4 Imagine that you are a taxi passenger. You have to pay the starting fee Rp and it charge Rp /km ……. Starting fee1 km2 km3 km4 km The Pattern of Sequence and Series Number

Adaptif Hal : 5 THE PROGRESSIONS Hal.: 5 SIGMA NOTATION The Concept of Sigma Notation Look at the sum of the first sixth odd number below: ……….. (1) In the form(1) The 1 st term = 1= 2.1 – 1 The 2 nd term= 3= 2.2 – 1 The 3 rd term = 5= 2.3 – 1 The 4 th term = 7= 2.4 – 1 The 5 th term = 9= 2.5 – 1 The 6 th term = 11= 2.6 – 1 Generally, the k-th term in (1) can be stated in the form of 2k – 1, k  { 1, 2, 3, 4, 5, 6 }

Adaptif Hal : 6 THE PROGRESSIONS Hal.: 6 SIGMA NOTATION In Sigma notation, the addition form (1) can be written as:

Adaptif Hal : 7 THE PROGRESSIONS Hal.: 7 In the form of It is read “sigma 2k – 1 from k =1 to 6” or “the sum of 2k – 1 for k = 1 sd k = 6” 1 is called lower limit and 6 is called upper limit, k is called index (some people called it variable) SIGMA NOTATION

Adaptif Hal : 8 THE PROGRESSIONS Hal.: 8 SIGMA NOTATION Generally

Adaptif Hal : 9 THE PROGRESSIONS Hal.: 9 Stated into sigma form 1. a + a 2 b + a 3 b 2 + a 4 b 3 + … + a 10 b 9 Example: Define the value of SIGMA NOTATION

Adaptif Hal : 10 THE PROGRESSIONS Hal.: 10 SIGMA NOTATION 2. (a + b) n =

Adaptif Hal : 11 THE PROGRESSIONS Hal.: 11 The properties of sigma notation :, For every integer a, b and n SIGMA NOTATION

Adaptif Hal : 12 THE PROGRESSIONS Hal.: 12 SIGMA NOTATION Example 1: Show that Answer :

Adaptif Hal : 13 THE PROGRESSIONS Hal.: 13 SIGMA NOTATION Define the value of Example 2 : Answer: = 6 ( ) = 6 ( ) = 6.91 = 546

Adaptif Hal : 14 THE PROGRESSIONS Hal.: 14 ARITHMETIC SEQUENCE AND SERIES  The orderly numbers like in speedometer have the same difference for every two orderly term, so it forms a sequence  Arithmetic sequence is sequence with difference two orderly term constant  The general form is : U 1, U 2, U 3, …., U n a, a + b, a + 2b,…., a + (n-1)b  In arithmetic sequence, we have U n – U n-1 = b, so U n = U n-1 + b

Adaptif Hal : 15 THE PROGRESSIONS Hal.: 15 If you start arithmetic sequence with the first term a and difference b, then you will get this following sequence The n-th term of arithmetic sequence is U n = a + ( n – 1 )b Where U n = n-th term a = the first term b = difference n = the term’s quantity ARITHMETIC SEQUENCE AND SERIES a a + b a + 2ba + 3b….a + (n-1)b

Adaptif Hal : 16 THE PROGRESSIONS Hal.: 16 If every term of arithmetic sequence is added, then we will get arithmetic series. Arithmetic series is the sum of terms of arithmetic sequence General form : U 1 + U 2 + U 3 + … + U n atau a + (a +b) + (a+2b) +… + (a+(n-1)b) The formula of the sum of the first term in arithmetic series is Where S = the sum of n-th term n = the quantity of term a = the first term b = difference = n-th term ARITHMETIC SEQUENCE AND SERIES

Adaptif Hal : 17 THE PROGRESSIONS Hal.: 17 Known: the sequence of 5, -2, -9, -16,…., find: a.The formula of n-th term b.The 25 th term Answer: The difference of two orderly terms in sequence 5,-2, -9,-16,…is constant, b= -7, so that the sequence is an arithmetic sequence a.The formula of the n-th term in arithmetic sequence is U n = 5 + ( n – 1 ). -7 U n = n + 7 U n = -7n + 12 b. The 25 th term of arithmetic sequence is : U 12 = = ARITHMETIC SEQUENCE AND SERIES

Adaptif Hal : 18 THE PROGRESSIONS Hal.: 18 Geometric sequence is a sequence which has the constant ratio between two orderly term There is blue paper. It will cut into two pieces GEOMETRIC SEQUENCE AND SERIES

Adaptif Hal : 19 THE PROGRESSIONS Hal.: 19 Look at the paper part that form a sequence Every two orderly terms of the sequence have the same ratio It seems that the ratio of every two orderly terms in the sequence is always constant. The sequence like this is called geometric sequence and the comparison of every two orderly term is called ratio (r) 12 4 U1U2 U3

Adaptif Hal : 20 THE PROGRESSIONS Hal.: 20 Geometric sequence is a sequence which have constant ratio for two orderly term General form: U 1, U 2, U 3, …., U n atau a, ar, ar 2, …., ar n-1 In geometric sequence If you start the geometric sequence with the first term a and the ratio is r, then you get the following sequence GEOMETRIC SEQUENCE AND SERIES

Adaptif Hal : 21 THE PROGRESSIONS Hal.: 21 The n-th term of geometric sequence is : GEOMETRIC SEQUENCE AND SERIES Start With the first term a Multiply with ratio r Write the multiplication result

Adaptif Hal : 22 THE PROGRESSIONS Hal.: 22 GEOMETRIC SEQUENCE AND SERIES The relation of terms in geometric sequence Like in arithmetic sequence, the relation between terms in geometric sequence can be explained as follows: Take U 12 as example : U 12 = a.r 11 U 12 = a.r 9.r 2 = U 10. r 2 U 12 = a.r 8.r 3 = U 9. r 3 U 12 = a.r 4.r 7 = U 5. r 7 U 12 = a.r 3.r 8 = U 4.r 8 Generally, it can be formulated U n = U k. r n-k

Adaptif Hal : 23 THE PROGRESSIONS Hal.: 23 GEOMETRIC SEQUENCE AND SERIES Geometric series is the sum of terms in geometric sequence General form U 1 + U 2 + U 3 + …. + U n a + ar + ar 2 + ….+ ar n-1 The formula of the n sum of the first term in geometric series is

Adaptif Hal : 24 THE PROGRESSIONS Hal.: 24 GEOMETRIC SEQUENCE AND SERIES Known sequence 27, 9, 3, 1, …..find a.The formula of the n-th term b. The 8 th term Answer: The ratio of two orderly terms in sequence 27,9,3, 1, …is constant, so that the sequence is a geometric sequence a. The formula of the n-th term in geometric sequence is

Adaptif Hal : 25 THE PROGRESSIONS GEOMETRIC SEQUENCE AND SERIES b. The 8th term of geometric sequence is

Adaptif Hal : 26 THE PROGRESSIONS Hal.: 26 Infinite geometric series is a geometric series which has infinite terms. If infinite geometric series is -1 < r < 1, then the sum of geometric series has sum limit (convergent). For n = ∞, r n is close to 0 So S ∞ = With S ∞ = the sum of infinite geometric series a = the first term r = ratio If r < -1 or r > 1, then the infinite geometric series will be divergent, means the sum of terms is not limited Infinite Geometric Series GEOMETRIC SEQUENCE AND SERIES

Adaptif Hal : 27 THE PROGRESSIONS Hal.: Find the sum of infinite geometric series : ….. Example : GEOMETRIC SEQUENCE AND SERIES Answer : a = 18 ;

Adaptif Hal : 28 THE PROGRESSIONS Hal.: An elastic ball is drop from the height of 2m. Every time it bounce from the floor, it has ¾ of the previous height. How long is the route that will be passed by the ball until it stop? GEOMETRIC SEQUENCE AND SERIES Look at the picture! The ball is drop from A, so AB is passed only once. Then CD, EF, etc is passed twice. The route is in geometric series with a = 3 and r = ¾ the length of the route is= 2 S ∞ - a = 14 So, the route length that pass by the ball is 14 m

Adaptif Hal : 29 THE PROGRESSIONS Hal.: 29