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Page 1: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Bell Ringer

Page 2: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Sequences and SeriesChapter 10, Section 1 (McGraw Precal)

Pages 590 – 594

Lesson Objective4.1A, 4.1B

The students will investigate several different types of sequences. Students will use Sigma Notation to represent and calculate sums of series.

Demonstration of Learning

Given 5 problems involving series, the students will correctly solve at least 3 of them.

Page 3: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

AP Calculus BC Learning Objectives

LO Students will be able to

4.1A Determine whether a series converges or diverges.

4.1B Determine or estimate the sum of a series.

Page 4: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

REVIEW of Pre CALCULUSSequences

Series

Page 5: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

SequenceA sequence is a list of things (usually numbers) that arein order.

3, 5, 7, 9, …

2nd term

1st term 3rd term

4th termThree dots means it goes on forever

Page 6: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

When the sequence goes on forever it is called an INFINITE sequence, otherwise it is a FINITE sequence

{1, 2, 3, 4, … }

{20, 25, 30, 35, … }

{1, 3, 5, 7}

{4, 3, 2, 1}

Let’s look at some sequences

What do you

notice?

Page 7: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Example 1aFind the next four terms of the sequence

{2, 7, 12, 17, …}

Since the nth term is not given, we must look

for a possible pattern

We can look at the difference between

each term

One possible pattern is that each term is 5 greater than the previous term

Page 8: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Example 1aFind the next four terms of the sequence

{2, 7, 12, 17, …}

5𝑡ℎ 𝑡𝑒𝑟𝑚 = 17 + 5

6𝑡ℎ 𝑡𝑒𝑟𝑚 = 17 + 10

7𝑡ℎ 𝑡𝑒𝑟𝑚 = 17 + 15

8𝑡ℎ 𝑡𝑒𝑟𝑚 = 17 + 20

{2, 7, 12, 17, 22, 27, 32, 37,… }

= 22

= 27

= 32

= 37

Page 9: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Example 1bFind the next four terms of the sequence

{2, 5, 10, 17, …}

Since the nth term is not given, we must look

for a possible pattern

We can look at the sum of the difference between terms.

One possible pattern is that each sum of the difference is equal to the odd sequence {3, 5, 7, …}

Page 10: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Example 1bFind the next four terms of the sequence

{2, 5, 10, 17, …}

𝑎2 − 𝑎1 = 5 − 2

𝑎3 − 𝑎2 = 10 − 5

𝑎4 − 𝑎3 = 17 − 10

{3, 5, 7, … }

= 3

= 5

= 7

This new sequence is our pattern

between terms We can

assume the next 4 in the

sequence

9,11,13,15

Page 11: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Example 1aFind the next four terms of the sequence

{2, 5, 10, 17, …}

{3, 5, 7,9,11,13,15… }

5𝑡ℎ 𝑡𝑒𝑟𝑚 = 17 + 9

6𝑡ℎ 𝑡𝑒𝑟𝑚 = 26 + 11

7𝑡ℎ 𝑡𝑒𝑟𝑚 = 37 + 13

8𝑡ℎ 𝑡𝑒𝑟𝑚 = 50 + 15

= 𝟐𝟔

= 𝟑𝟕

= 𝟓𝟎

= 𝟔𝟓

{2, 7, 12, 17, 26, 37, 50, 65,… }

Page 12: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Example 1cFind the first four terms of the sequence

𝑎𝑛 = 2𝑛 −1𝑛

Since the nth term is given, we can just plug and chug!!

Page 13: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Example 1cFind the first four terms of the sequence

𝑎𝑛 = 2𝑛 −1𝑛

𝑎1 = 2 1 −11

𝑎2 = 2 2 −12

𝑎3 = 2 3 −13

{−2, 4, −6, 8… }

= −2

= 4

= −6

𝑎4 = 2 4 −14 = 8

𝑛 = 1

𝑛 = 2

𝑛 = 3

𝑛 = 4

Page 14: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Explicit versus Recursive

Explicit – Infinitely many sequences exist with the

same first few terms. To sufficiently define a

unique sequence, a formula for the nth term or

other information must be given. An explicit

formula gives the nth term an as a function of n.

Recursive – Recursively defined sequences give

one or more of the first few terms and then define

the terms that follow using those previous terms.

Page 15: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Example 2a

Find the fifth term of the recursively defined

sequence 𝑎1 = 2, 𝑎𝑛 = 𝑎𝑛−1 + 2𝑛 − 1 where, 𝑛 ≥ 2

Since the sequence is defined recursively, all the terms before the fifth term must be found first.

Page 16: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Example 2a

Find the fifth term of the recursively defined

sequence 𝑎1 = 2, 𝑎𝑛 = 𝑎𝑛−1 + 2𝑛 − 1 where, 𝑛 ≥ 2

𝑎2 = 𝑎2−1 + 2 2 − 1 𝑎2 = 5𝑛 = 2= 𝑎1 + 3

= 2 + 3

𝑎3 = 𝑎3−1 + 2 3 − 1 𝑎3 = 10𝑛 = 3

= 𝑎2 + 5

= 5 + 5𝑎4 = 𝑎4−1 + 2 4 − 1 𝑎4 = 17𝑛 = 4

= 𝑎3 + 7

= 10 + 7

Page 17: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Example 2a

Find the fifth term of the recursively defined

sequence 𝑎1 = 2, 𝑎𝑛 = 𝑎𝑛−1 + 2𝑛 − 1 where, 𝑛 ≥ 2

𝑎5 = 𝑎5−1 + 2 5 − 1 𝑎5 = 26𝑛 = 5= 𝑎4 + 9

= 17 + 9

Page 18: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Convergent versus Divergent

Convergent – A sequence whose limit approaches

a unique number.

Divergent – A sequence whose limit DOES NOT

approaches a unique number.

Page 19: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Example 3a

Determine whether the sequence is convergent or

divergent

𝑎𝑛 = −3𝑛 + 12

𝑎0 = −3 0 + 12 𝑎0 = 12𝑛 = 0𝑎1 = −3 1 + 12 𝑎1 = 9𝑛 = 1𝑎2 = −3 2 + 12 𝑎2 = 6𝑛 = 2

𝑎3 = −3 3 + 12 𝑎3 = 3𝑛 = 3𝑎4 = −3 4 + 12 𝑎4 = 0𝑛 = 4𝑎5 = −3 5 + 12 𝑎5 = −3𝑛 = 5𝑎6 = −3 6 + 12 𝑎6 = −6𝑛 = 6𝑎7 = −3 7 + 12 𝑎7 = −9𝑛 = 7

Page 20: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Example 3a

Determine whether the sequence is convergent or

divergent

𝑎𝑛 = −3𝑛 + 12

𝑎0 = 12𝑎1 = 9𝑎2 = 6

𝑎3 = 3𝑎4 = 0𝑎5 = −3𝑎6 = −6𝑎7 = −9

2 4 6 8 10 n

4

8

-4

-8

an

Page 21: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Example 3a

Determine whether the sequence is convergent or

divergent

𝑎𝑛 = −3𝑛 + 12

2 4 6 8 10 n

4

8

-4

-8

an

an does NOT

approach a finite

number, so it must be divergent

Page 22: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Example 3b

Determine whether the sequence is convergent or

divergent

𝑎1 = 36𝑛 = 1𝑎2 = −

1

236 𝑎2 = −18𝑛 = 2

𝑎3 = −1

2−18 𝑎3 = 9𝑛 = 3

𝑎4 = −1

29 𝑎4 = −4.5𝑛 = 4

𝑎5 = −1

2−4.5 𝑎5 = 2.25𝑛 = 5

𝑎6 = −1

22.25 𝑎6 = −1.125𝑛 = 6

𝑎7 = −1

2−1.125 𝑎7 = 0.5625𝑛 = 7

𝑎8 = −1

20.5625 𝑎8 = −0.2813𝑛 = 8

𝑎𝑛 = 36, 𝑎𝑛 = −1

2𝑎𝑛−1, 𝑎 ≥ 2

Page 23: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Example 3b

Determine whether the sequence is convergent or

divergent

2 4 6 8 10 n

24

36

-12

an

𝑎1 = 36𝑎2 = −18𝑎3 = 9𝑎4 = −4.5

𝑎5 = 2.25

𝑎6 = −1.125

𝑎7 = 0.5625

𝑎8 = −0.2813

12

𝑎𝑛 = 36, 𝑎𝑛 = −1

2𝑎𝑛−1, 𝑎 ≥ 2

Page 24: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Example 3b

Determine whether the sequence is convergent or

divergent

2 4 6 8 10 n

24

36

-12

an

12

𝑎𝑛 = 36, 𝑎𝑛 = −1

2𝑎𝑛−1, 𝑎 ≥ 2

an does approach a

finite number, so it must be

convergent𝐥𝐢𝐦𝒏→∞𝒂𝒏 = 𝟎

an does approach a

finite number, so it must be

convergent

Page 25: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Example 3c

Determine whether the sequence is convergent or

divergent

𝑎1 = −0.2 𝑎2 = 0.222

𝑎3 = −0.231 𝑎4 = 0.235

𝑎5 = −0.238 𝑎6 = 0.24

𝑎7 = −0.241 𝑎8 = 0.242

𝑎𝑛 =−1 𝑛 ∙ 𝑛

4𝑛 + 1

𝑎9 = −0.243 𝑎10 = 0.244

𝑎11 = −0.244 𝑎12 = 0.245

What pattern do

you notice?

Page 26: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Example 3c

Determine whether the sequence is convergent or

divergent

2 4 6 8 10 n

0.25

an

𝑎𝑛 =−1 𝑛 ∙ 𝑛

4𝑛 + 1

-0.25

an

approaches 0.25 when n is even, and -0.25 when n is

odd.

an does NOT

approach a unique finite

number, so it must be divergent

Page 27: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

SeriesA series is the indicated sum of all the terms of asequence.

{𝟑, 𝟓, 𝟕, 𝟗, … } 𝟑 + 𝟓 + 𝟕 + 𝟗 +⋯=

When the series is a sum of an infinite sequence, we call it an INFINITE series; when it is the sum of a finite

sequence, we call it a FINITE series.

Page 28: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Example 4aFind the fourth Partial Sum of

We must find the first four terms.

Then find the sum of the first four terms.

𝑎𝑛 = −2𝑛 + 3

nth Partial SumAn nth partial sum is the sum of the first n terms and isdenoted by 𝑆𝑛

Page 29: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Example 4aFind the fourth Partial Sum of

𝑎𝑛 = −2𝑛 + 3

𝑎1 = (−2)1 + 3 𝑎1 = 1𝑛 = 1

𝑎2 = 7𝑛 = 2𝑎3 = −5𝑛 = 3𝑎4 = 19𝑛 = 4

𝑎2 = (−2)2 + 3

𝑎3 = (−2)3 + 3

𝑎4 = (−2)4 + 3

𝑆4 = 1 + 7 + −5 + 19

𝑆4 = 22

Page 30: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Example 4b

Find 𝑆3 of 𝑎𝑛 =4

10𝑛

𝑎1 =4

101𝑎1 = 0.4𝑛 = 1

𝑎2 = 0.04𝑛 = 2

𝑎3 = 0.004𝑛 = 3

𝑎2 =4

102

𝑎3 =4

103

𝑆3 = 0.4 + 0.04 + 0.004

𝑆3 = 0.444

Page 31: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Example 5aFind the sum of

Sigma NotationTo denote a series, we can use this notation:

𝑛=1

𝑘

𝑎𝑛 = 𝑎1 + 𝑎2 + 𝑎3 +⋯+ 𝑎𝑘

Where n is the index of summation, k is the upperbound of summation, and 1 is the lower bound ofsummation.

𝑛=1

5

4𝑛 − 3

Page 32: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Example 5aFind the sum of

𝑛=1

5

4𝑛 − 3

4(1) − 3 = 1𝑛 = 1= 5𝑛 = 2= 9𝑛 = 3= 13𝑛 = 4

4(2) − 34(3) − 34(4) − 3

𝑛=1

5

4𝑛 − 3 = 1 + 5 + 9 + 13 + 17

= 17𝑛 = 5 4(5) − 3

= 45

Page 33: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Example 5bFind the sum of

𝑛=3

76n − 3

2

6 3 − 3

2= 7.5𝑛 = 3

= 10.5𝑛 = 4

= 13.5𝑛 = 5

= 16.5𝑛 = 76 4 − 3

2

6 7 − 3

2

6 5 − 3

2

𝑛=3

76n − 3

2= 7.5 + 10.5 + 13.5 + 16.5 + 19.5

= 19.5

𝑛 = 6 6 6 − 3

2

= 67.5

Page 34: Bell Ringer · 2019-10-17 · Explicit versus Recursive Explicit –Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula

Example 5cFind the sum of

𝑛=1

∞7

10n

7

101= 0.7𝑛 = 1

= 0.07𝑛 = 2

= 0.007𝑛 = 3

= 0.0007𝑛 = 57

1027

105

7

103

𝑛=1

∞7

10n= 0.7 + 0.07 + 0.007 + 0.0007 + 0.00007 +⋯

= 0.00007

𝑛 = 4 7

104

= 0.77777… . 𝑜𝑟7

9