Sequence and Series What is the significance of sequence and series?

Sequence and Series General Patterns What is the difference between a sequence and a series?

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SEQUENCE A set of numbers which follows a pattern 4

TERMS The individual parts of the sequence 5

INFINITE SEQUENCE A sequence with no last term 6

EXPLICIT SEQUENCE A sequence found by substituting the position of the term into a formula. 7

Find the 5 th and 7 th terms of the sequence Given: a n = 2 n – 1 5 th : a 5 = 2 5 – 1 = 7 th : a 7 = 2 7 – 1 = Determine the Sequence from the formula

Find the 4 th, and 10 th terms Given: a n = 3n-1 4 th : a 4 = 3(4) – 1 = 10 th : a 10 = 3(10) – 1 =

Find the general term for the sequence 2, 4, 8, 16, 32, …. Position n= 1 st 2 nd 3 rd 4 th 5 th Always ask how is n related to the item in the sequence? often the relationship is a multiple of n: or the term is n squared or cubed, etc.: or the term indicates that some number is raised to the n th power. What is the relationship for the sequence above? a n = 2 n 10 Find the formula from the sequence

Find the general term for the sequence -1, 2, -3, 4, -5, 6… Position n= 1 st 2 nd 3 rd 4 th 5 th Notice how the sign changes: what can make it change? Raising -1 to a power (-1) n or n+1 use n when the negative is on the odd terms n+1 when the negative is on the even terms 11

RECURSIVE SEQUENCE A sequence where each term uses the previous term or terms to find the next one. 12

Find the first five terms of the recursive sequence where a 1 = 1 and a n = 3a n-1 – 1 Always has two parts, the 1 st term and a rule to follow a 1 = 1 a 2 = 3(1) – 1 = a 3 = 3(2) – 1 = a 4 = 3(5) – 1 = a 5 = 3(14) – 1 =

Find the first five terms of the recursive sequence where a 1 = 2, a 2 = 2 and a n = a n-1 – a n-2 a 1 = 2 a 2 = 2 a 3 = 2 – 2 = a 4 = 0 – 2 = a 5 = -2 – 0 =

SERIES The sum of a specific number of terms of the sequence S n = a 1 + a 2 + a 3 + … + a n 15

Find S 2, S 3 and S 5 given the sequence -2, 4, -6, 8, -10, 12, -14 S 2 = a 1 + a 2 = = S 3 = a 1 + a 2 + a 3 = = S 5 = a 1 + a 2 + a 3 + a 4 + a 5 = =

4  (2n + 1) n = 1 17 Summation Notation   Means to sum up n = 1 n = 1 tells which number to substitute first 4 4 Tell the last number to be substituted (2n + 1) (2n + 1) the function you are to use

4  (2n + 1) n = 1 = (2(1)+1) + (2(2)+1) + (2(3)+1) + (2(4)+1) = = 24 18

Write Sigma notation for n= 1 st 2 nd 3 rd 4 th 5 th Find the pattern! state the ending point 5 State the rule  2n state the starting point n = 1 19

Write Sigma notation for n = 1 st 2 nd 3 rd 4 th or n = 2 nd 3 rd 4 th 5 th Find the pattern! ∞  n = n21n2

homework Worksheet 1 21

Sequence and Series What makes a sequence arithmetic?

What pattern is formed when graphing an arithmetic sequence 3, 5, 7, 9,... 1 st 2 nd 3 rd 4 th Creates points: (1, 3), (2, 5), (3, 7). (4, 9) What do we know about lines? y = mx + b 23 Activation:

ARITHMETIC SEQUENCE is a sequence in which a constant,d, called the common difference, is “added” to each term to get the next term 24

Find the first term and common difference for the sequence: 2, 5, 8, 11, 14 First Term: 2 Common Difference: a 2 – a 1 5 – 2 = 3 25

Find the first term and common difference for the sequence: 20, 13, 6, -1, -8 First Term: 20Common Difference: -7 26

Find the first term and common difference for the sequence: 1. 3, 1, 5, , 4, 5, First Term: 1Common Difference:

N th TERM Developing The General Term Just watch If a 1 = a 1 Then a 2 =a 1 + d And a 3 =a 2 + d Or =(a 1 + d) + d =a 1 + 2d What would a 4 = d a 1 + 3d

N th TERM Or the General Formula for an Arithmetic Sequence a n = a 1 + (n-1)d 29

Find the 14 th term of the arithmetic sequence 4, 7, 10, 13 … What do you need to know and what do you know? a 1 = 4 d = 3 n = 14 a n = a 1 + (n-1)d a 14 = 4 + (14 – 1)3 a 14 = a 14 = 43 30

Find the 20 th term of the arithmetic sequence 7, 4, 1 … a 1 = 7 d = -3 n = 20 a n = a 1 + (n-1)d a 20 = 7 + (20 – 1)-3 a 20 = a 20 =

In the sequence 4, 7, 10, 13 …, which term has a value of 301? What do we know? a1 a1 = 4 d = 3 n = ? an an = a n = a1 a1 + (n – 1)d 301 = 4 + (n – 1)3 301 = 4 + 3n = 3n 100 = n 301 is the 100 th term

In the sequence 2, 6, 10, 14 …, which term has a value of 286? What do we know? a 1 = 2 d = 4 n = ? a n = 286 a n = a 1 + (n – 1)d 286 = 2 + (n – 1)4 286 = 2 + 4n = 4n 72 = n284 is the 72 nd term 33

The 3 rd term is 8 and the 16 th term is 47. Find a 1 and d and construct the sequence. (3, 8) For a 3 = 8 8 = a 1 + (3-1)d 8 = a 1 + 2d (16, 47) For a 16 = = a 1 + (16-1)d 47 = a d Provides two equations two unknowns a d = 47 a 1 + 2d = 8 13d = 39 d = 3 Substitute a 1 +2(3) = 8 a = 8 a 1 = 2 34 Write the Sequence: 2, 5, 8, 11, 14, … -( )

The 3 rd term is 8 and the 16 th term is 47. Find a 1 and d and construct the sequence. 35 Write the Sequence: 2, 5, 8, 11, 14, … d= 47 – 8 = – 3 13 d=3 Still need one equation: a 3 = a 1 + 2d 8= a 1 + 2(3) 8 = a = a 1 Since we know that an arithmetic sequence is linear you could use this Alternate method

The 7 th term is 79 and the 13 th term is 151. Find a 1 and d and construct the sequence. d = 151 – 79 = – 7 6 d = 12 a 1 +2(12) = 79 a = 79 a 1 = 7 36 Sequence: 7, 19, 31, 43, 55, …

ARITHMETIC MEAN Values inserted between two numbers a and b such that an arithmetic sequence is formed 37

Insert 3 arithmetic means between 8 and 16. 8, ____, ____, ____, 16 a 1 a 2 a 3 a 4 a 5 Therefore 8 is the 1 st term and 16 is the 5 th term. a n = a 1 + (n-1)d a 5 = a 1 + (5-1)d 16 = 8 + (5-1)d 16 = 8 + 4d 8 = 4d 2 = d

Insert 2 arithmetic means between 3 and 24. 3, ____, ____, 24 a 1 a 4 24 = 3 + (4-1)d 24 = 3 + 3d 21 = 3d 7 = d Sequence: 3, 10, 17, 24 39

Arithmetic Series Formula S n = (a 1 + a n ) used when you have 1 st and last terms OR Since a n = a 1 + (n-1) d = (a 1 + a 1 + (n-1) d) S n = (2 a 1 + (n-1) d) used when you have 1st and common difference 40 n2n2 n2n2 n2n2

Find the sum of the first 100 natural numbers … What do we know? a 1 = 1a n = 100 n = 100 S n = (a 1 + a n ) S 100 = ( ) = 50(101) = n2n

Find the sum of the first 14 terms of … What do we know? a 1 = 2d = 3 n = 14 S n = (2a 1 + (n-1)d) S 14 = (2(2) + (14-1)3) = 7(4 + (13)3) = 7(43) = n2n2 14 2

Find the sum of the series 13  (4n + 5) n = 1 S 13 = (2(9) + (13-1)4) = (18 + (12)4) = (66) = Find a few terms so you can tell it is arithmetic … a 1 = 9 d = 4 n =

Find the sum of the series 10  (9n - 4) n = 1 S 10 = (2(5) + (10-1)9) = 5(10 + (9)9) = 5(91) = Find a few terms … a 1 = 5 d = 9 n =

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Sequence and Series What makes a sequence or series geometric?

What pattern is formed when graphing a geometric sequence 2, 6, 18, 54,.. 1 st 2 nd 3 rd 4 th Creates points: (1, 2), (2, 6), (3, 18). (4, 54) What pattern do you see? y = a x 47 Activation:

GEOMETRIC SEQUENCE Is a sequence in which a constant, r, the common ratio, can be “multiplied” by each term to get the next term 48

Find the common ratio for the geometric sequence: 3, 6, 12, 24,… r =a 2 ÷ a 1 =6 ÷ 3 =2 Common Ratio: 2 It will always be the same so you only need to check one if you know it is geometric. 49

Find the common ratio for the geometric sequence: 1, ½, ¼, 1/8,… a 1 = 1 a 2 = ½ a 2 ÷ a 1 ½ ÷ 1 ½ Common Ratio: ½ 50

Geometric sequence formula Just watch If a 1 = a 1 Then a 2 = a 1 r And a 3 = a 2 r Or = (a 1 r)r = a 1 r 2 a 4 = 51 r0r0 1 a 1 r 3

Geometric sequence formula What do you think a n will be? a n = a 1 r n-1 52

Find the 6 th term of the geometric sequence: 3, -15, 75… What do we know? a 1 = 3 n = 6 r = = -5 a n = a 1 r n-1 a 6 = 3(-5) 6-1 = 3(-5) 5 = 3(-3125) =

Find the 10 th term of the geometric sequence: 8, 4, 2 … a 1 = 8 n = 10 r = 4 ÷ 8 = 4 x 243 = a n = a 1 r n-1 = =

GEOMETRIC MEAN Values inserted between two numbers a and b such that a geometric sequence is formed 55

Insert 2 geometric means between 3 and 24. 3, ____, ____, 24 3 is the first term and 24 is the 4 th term. a 4 = a 1 r (4-1) 24 = 3r( 4-1) 8 = r 3 2 = r Sequence: 3, 6, 12, 24 56

Insert 3 geometric means between ¼ and 1/64. ¼ is the first term and 1/64 is the 5 th term. 1/64 = ¼ r 5-1 1/16 = r 4 ± ½ = r Sequence: ¼, ± 1/8, 1/16, ± 1/32, 1/64 57

GEOMETRIC SERIES The sum of a geometric sequence S n = a 1 (1-r n ) 1-r 58

Find the sum of the first 6 terms of … What do we know? a 1 = 3n = 6 r = = 2 S n = a 1 (1-r n ) 1-r S 6 = 3(1-2 6 ) = 3(1-64) = 3(-63) =

Find the sum of the first 10 terms of 2 – 1 + ½ - ¼ + … What do we know? a 1 = 2n = 10 r = S n = a 1 (1-r n ) 1-r S 10 = 2(1-(-1/2) 10 ) = 2(1-1/1024) 1-(-1/2) 3/2 = 2(1023/1024) = 1023/512 = 341/256 3/2 3/2 60 2

Find the sum of the geometric series 6  3 n n = 1 S 6 = 3 (1 – (3) 6 ) 1- 3 = 3 (1 – 729) = 3 (-728) = = Find the few terms to verify that the sequence is geometric … … a 1 = 3 n = 6 r = 3

Find the sum of the geometric series 5  (½) n+1 n = 1 S 5 = ¼ (1 – (½) 5 ) 1- ½ = ¼ (1 – (1/32)) = ¼ (31/32) = ¼ (31/32) 2 ½ ½ = 31/64 62 Find the few terms. (½) 2 + (½) 3 + (½) 4 + … a 1 = ¼ n = 5 r = ½

A student borrows $600 at 12% interest compounded monthly. The student pays the loan in one payment at the end of the 36 months. How much does the student pay? Annual Interest: 12% = 0.12 Monthly Interest: 1% = 0.01 Find the pattern!! 1 st Month: P 2 nd Month: P rd Month: P1.01(1.01) 4 th Month: P1.01 (1.01) (1.01) N th Month: P(1.01) n-1 a n = 600 n = 37 since P is 1 st month r = 1.01 a 37 = 600(1.01) 37-1 a 37 = 600(1.01) 36 a 37 = 600( ) a 37 = $

Try the problem with the compound interest formula: 64 A student borrows $600 at 12% interest compounded monthly. The student pays the loan in one payment at the end of the 36 months. How much does the student pay?

homework Worksheet 3 65

Infinite Geometric Series What is an infinite series?

Activation: Can an infinite series have a sum? Given: 20, 10, 5, … Find S 3 = S 5 = S 10 = 67 Given: 3, 6, 12,... Find S 3 = S 5 = S 10 =

INFINITE GEOMETRIC SERIES A series associated with a geometric sequence that has no last term 68

CONVERGENT Converge—to move towards something Convergent Series—moves towards a particular number and has a limit (1 st intro to Calculus) 69

DIVERGENT Diverge—to move away from something A Divergent series—gets larger or smaller without bound and has no limit. 70

Convergent or Divergent? How do you know? if |r|<1. the series is convergent 71

Determine which geometric series have sums A.1 – ½ + ¼ - … r = - ½ |r| < 1 Series has a sum B … r = 5 |r| > 1 Series does not have a sum C. 1 + (-1) … r = (-1) |r| = 1 Series does not have a sum 72

Finding The sum of an Infinite series 73

Find the sum of the infinite geometric series 5 + 5/2 + 5/4 + 5/8 + … a 1 = 5 r = ½ S = 5 = 5 = 5 x 2 = 10 1 – ½ ½ 74

Find the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + … a 1 = 1 r = 1/3 S = 1 = 1 = 3 1 – 1/3 2/3 2 75

Find the sum of the repeating decimal 0.6 also represented as , … a 1 = 0.6 r = 0.1 S = 0.6 = 0.6 = 2 1 –

homework Infinite Series 77

Scavenger Hunt Rules You must stay with your group at all times! You must remain quiet in the hallways. You need to show work in order to get credit. You should return to the classroom 5 min before the end of the period. This is worth 10 points towards Classwork. First team back wins an additional prize. 78

homework Review Sheet 79