Sequences, Series and the Binomial Theorem

Independent Study Unit

Due:_______________ ISU Outline

1. Sequences and Series 2-3. Arithmetic and Geometric Sequences 4-5. Arithmetic and Geometric Series 6. Recursion – Fibonacci Numbers, Koch’s snowflake, Sierpinski’s triangle 7. Binomial Theorem – Summation Notation, Sigma, Pascal’s Triangle 8. Assignment

Instructions: The following is designed as an Independent Study Unit. The dates on this page indicate the class that has been set aside to work on each topic. It is your responsibility to view the example video and attempt the follow-up questions for each topic before the dates shown on this page. ���� Important terminology or ideas will be bolded. There can be three sections with each topic: ���� Your turn: These are questions for you to try. Solutions will be given so that you can see if you are on the right track. ���� Consolidation: These are questions designed to give you more practice. ���� Extension: These are questions or activities to extend your knowledge on the topic.

Independent Study Unit:

Date Topic Homework

Wednesday Dec. 16 1 Intro to Sequences and Series Pg. 433 # 1 – 4

Thursday Dec. 17 2 Arithmetic Sequences Pg. 441 # 1 - 10

Friday Dec. 18 Word Problems

Monday Jan. 4th

3 Geometric Sequences Pg. 452 # 1 -7, 26 - 28

Tuesday Jan. 5th

4 Arithmetic Series Pg.469 # 1 - 7, 19, 20, 23, 24

Wednesday Jan. 6th

5 Geometric Series Pg. 477 # 1 - 5

Thursday Jan. 7th

6 Recursion Pg. 461 # 1 - 4

Friday Jan. 8th

7 Sigma Notation, Pascal’s Triangle, Binomial Theorem WS

Monday Jan. 11th

8 Review Pg. 480 # 1 - 35

Tuesday Jan 12th

9 Unit Test

1. Sequences and Series Definitions: ���� A Sequence is a list (set) of numbers following a pattern in some kind of order. Example: a) 1, 2, 3, 4, 5, …..

b) 2, 4, 6, 8, … c) 2, 4, 8, 16, …. d) 0, 1, 4, 9, 16, …

���� Individual numbers in the list (set) are called terms and are identified with tn where n is the number or position of the term. Example: 1, 2, 3, 4, 5, ….. t1 t2 t3 t4 t5 … ���� A Series is the sum of the sequence. Example: a) 1 + 2 + 3 + 4 + 5 +…..

b) 2 + 4 + 6 + 8 +… c) 2 + 4 + 8 + 16 +…. d) 0 + 1 + 4 + 9 + 16 + …

���� Implicit Formula is an equation or formula to calculate the individual terms. Example: a) 1 + 2 + 3 + 4 + 5 +….. Formula: ntn =

b) 2 + 4 + 6 + 8 +… Formula: ntn 2=

c) 2 + 4 + 8 + 16 +…. Formula: nnt 2=

d) 0 + 1 + 4 + 9 + 16 + … Formula: 2ntn =

where n starts at 1 and increases by 1 each time. ���� Your turn: 1. Given the formula for the nth term state the first 5 terms and any other given terms:

a) ntn 25−= , 10150,tt b) 12 1 −= −nnt , 1510,tt c)

( )n

ntn

23−= , 2512,tt

2. Given the sequence find a formula for the nth term (if possible) and state the next 3 terms:

a) ,...5

4,

4

3,

3

2,

2

1 b) ,...19,15,10,6,1 c) ,...

35

,38

,3

10,

311

���� Consolidation:

HW. pg. 433 #1-4

2. Arithmetic Sequences

Watch the Khan Academy Videos on Arithmetic Sequences:

https://www.khanacademy.org/math/algebra/sequences/arithmetic_sequences/v/arithmetic-sequences (7 mins) ���� An Arithmetic Sequence is a list (set) of numbers with a Common Difference.

What is the common difference? Example: a) 2, 5, 8, 11, 14, … + 3 b) 5, 3, 1, … + (– 2) or – 2 c) 18, 44, 70, … + 26

d) ,...1,3

2,

3

1 +

3

1

���� Hint: to find a “common difference” or “d” choose a term and subtract the previous term.

12 ttd −= or in general 1−−= nn ttd

���� Note: All sequences have an initial term or a first term, t 1, which is usually indicated by the variable “a”. ���� Go back to the previous examples and indicate the first term either by writing it out or by circling it. ���� Implicit Formula – Using what we know: d, r, a, n Arithmetic Example:

Given a, d, n where a is the first term, d is the common difference and n is the term number

2, 5, 8, 11, 14, … a = 2, d = 3, n = ?

Formula: (General Term)

dnatn )1( −+=

Formula:

3)1(2 −+= ntn

332 −+= ntn

ntn 31+−=

���� Note: Once you have the general term or formula you can find any term or any missing info about a term.

���� Your turn:

1. Given the sequences: i) 2, 6, 10, … ii) ,...4,2

9,5

For each one: a) Write the next 3 terms. b) State a and d and give the formula for tn. c) Find the 12th term and t22. d) How many terms would there be (ie. what is the term number) if the sequence ended at: i) 62 ii) 0

9. Suppose two terms of an Arithmetic sequence are t3 = 11 and t9 = 29. Find the formula and the first 3 terms in the sequence.

10. Given two terms of a Arithmetic sequence find nt .

a) 25,16 85 == tt b) 5,5.2 214 −== tt

���� Consolidation:

HW. pg. 441 #1-10

3. Geometric Sequences

Watch the Khan Academy Videos on Geometric Sequences:

https://www.khanacademy.org/math/algebra/sequences/geometric_sequences/v/geometric-sequences-introduction (11 mins)

���� A Geometric Sequence is a list (set) of numbers with a Common Ratio.

What is the common ratio? Example: a) 2, 4, 8, … 2×

b) 25, 5, 1, … 5

1× or 5÷

c) 3, -24, 192, … )8(−×

d) ,...34

,32

,31

2×

���� Hint: to find a “common ratio” or “r” choose a term and divide it by the previous term.

1

2

t

td = or in general

1−

=n

n

t

td

���� Note: All sequences have an initial term or a first term, t 1, which is usually indicated by the variable “a”. ���� Go back to the previous examples and indicate the first term either by writing it out or by circling it. ���� Implicit Formula – Using what we know: d, r, a, n

���� Note: Once you have the general term or formula you can find any term or any missing info about a term.

Geometric Example:

Given a, r, n where a is the first term, r is the common ratio and n is the term number

3, -24, 192, … a = 3, r = -8, n = ?

Formula: (General Term)

( )1−= nn rat

Formula:

( )1)8(3 −−= nnt

���� Your turn:

1. Given the sequences: i) 2, 6, 18, … ii) ,...4

5,

2

5,5

For each one: a) Write the next 3 terms. b) State a and d and give the formula for tn. c) Find the 12th term and t22. d) How many terms would there be (ie. what is the term number) if the sequence

ended at: i) 13122 ii) 128

5

2. Suppose two terms of a Geometric sequence are t3=250 and t8=781250. Find the formula and the first 3 terms in the sequence.

3. Given two terms of a Geometric sequence find nt .

a) 108,36 43 == tt b) 11,99 53 == tt

���� Consolidation:

HW. pg. 452 #1-7, 26-28 ���� Test Youself: 1. Determine whether the sequence is Arithmetic, Geometric or neither:

a) 1, 4, 9, 16, … b) 1, 2, 4, 8, … c) 32, 16, 8, 4, … d) 20, 16, 12, 8, … e) ,...3

5,

3

8,

3

10,

3

11

2. State a, r or d and the formula for nt , then find the indicated term.

a) 2, 4, 8, …, 7t and 12t b) 6, 0.6, 0.06, …, 8t and 15t c) 6, 8, 10, …, 10t and 24t

d) ,...2

5,

2

3,

2

112t and 21t

4. Arithmetic Series

Watch the Video on Arithmetic Series:

https://www.youtube.com/watch?v=Dj1JZIdIwwo (7 mins)

• An Arithmetic Sequence is a list(set) of numbers with a common difference, d.

Given a, d, n where a is the first term, d is the common difference and n is the term number.

General Term: dnatn )1( −+=

• An Arithmetic Series is the sum of the sequence.

Individual numbers in the series are still called terms and are identified with tn where n is the number or position of the term. The sum up to a particular term, tn, will now be identified with Sn, where Sn is the sum of the n terms. Sequence Series 2, 5, 8, 11, 14, … 2 + 5 + 8 + 11 + 14 + … Given a, d, n where a is the first term, d is the common difference and n is the term number.

Sum formula: [ ]dnan

Sn )1(22

−+=

Example: Given 5 + 9 + 13 +… Then a = 5, d = 4, n = 12

Find S12

[ ])4)(112()5(22

1212 −+=S

[ ])4)(11(10612 +=S

[ ]4410612 +=S

32412 =S So the sum of the first 12 terms in the series is 324.

���� Your turn: 1. Given the sequences:

i) ...42

95 +++ ii) 2 + 6 +18 + … iii) 2 + 6 + 10, … iv) – 3 + 6 – 12 + 24 … v) ...

4

5

2

55 +++

For each one: a) Decide if the series is Arithmetic or Geometric. b) Write the next 3 terms. c) For any Arithmetic Series state a and d and give the formula for tn and Sn. d) Find the sum of the first 12 terms and S22.

2. Given the series: 2 + 5 + 8 +… . If the last term is 101 what is the sum?

���� Consolidation:HW . pg. 469 #1-7, 19, 20,23,24

5. Geometric Series

Watch the Video on Arithmetic Series:

https://www.youtube.com/watch?v=CecgFWTg9pQ (7 mins)

• A Geometric Sequence is a list(set) of numbers with a common ratio, r. Given a, r, n where a is the first term, r is the common ratio and n is the term number.

General Term: ( )1−= nn rat

• A Geometric Series is the sum of the sequence.

Individual numbers in the series are still called terms and are identified with tn where n is the number or position of the term. The sum up to a particular term, tn, will now be identified with Sn, where Sn is the sum of the n terms. Sequence Series 2x, 4x2, 8x3, … 2x + 4x2 + 8x3 +… Given a, r, n where a is the first term, r is the common ratio and n is the term number.

Sum Formula: ( )

11

−−=

r

raS

n

n

Example: Given 5 + 20 + 80 +… Then a = 5, r = 4, n = 7

Find S7

( )14

145 7

−−=nS

( )3

1163845 −=nS

27305=nS

So the sum of the first 7 terms in the series is 27305. ���� Your turn: 1. Given the sequences:

i) ...429

5 +++ ii) 2 + 6 +18 + … iii) 2 + 6 + 10, … iv) – 3 + 6 – 12 + 24 … v) ...45

25

5 +++

For each one: e) Decide if the series is Arithmetic or Geometric. f) Write the next 3 terms. g) For any Geometric Series state a and r and give the formula for tn and Sn. h) Find the sum of the first 12 terms and S22.

2. Find the sum of the series 5 – 15 + 45 – … + 3645.

3. Suppose two terms of a Geometric Series are 21

1 =t and 31

2 =t , find S6 and the formula for the Sn.

���� Consolidation:HW . pg. 477 #1-6,

���� Extension: 1. The story of “little” Gauss:

Watch the Numberphile Video on Karl Gauss:

https://www.youtube.com/watch?v=Dd81F6-Ar_0 (7 mins)

2. The first three terms of an arithmetic series have a sum of 24 and a product of 312. What is the fourth terms of the series? 3. Determine the first and last terms of an arithmetic series with 50 terms, a common difference of 6 and a sum of 7850. 4. A geometric series has three terms. The sum of the three terms is 42. The third term is 3.2 times the sum of the other two terms. What are the terms? 5. A ball is thrown 16m into the air. The ball falls, rebounds to half its previous height, and falls again. If the ball continues to rebound and fall in this manner, finds the total distance the ball travels until it hits the ground for the sixth time.

6. Recursion ���� Recursion occurs when later terms are based/calculated from previous terms.

Example: Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, 21, …

Can you see how too get the next term? Hint: 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13, …

In general terms: 12 −− += nnn ttt

You can see that you need all the previous terms to calculate any new terms.

Watch the videos on the Fibonacci Numbers:

https://www.youtube.com/watch?v=P0tLbl5LrJ8 (4 mins) https://www.ted.com/talks/arthur_benjamin_the_magic_of_fibonacci_numbers?language=en (6 mins) ���� Consolidation: 1. Given the formula for the nth term state the first 5 terms: a) 2

12 ntt nn += − , 21 −=t b) 122 −− −= nnn ttt , 1,1 21 −== tt c) 213 32 +++ −+= nnnn tttt ,

1,5,12 321 =−== ttt

2. Write an explicit formula for the nth term for the following:

a) 21 −= −nn tt , 101 =t b) 12

1−= nn tt , 201 =t c)

)1(

11 +

+= − nntt nn ,

2

11 =t

���� Extension: 1. Look up Koch’s snowflake and Sierpinski’s Triangle on the Internet. Explain how these are examples of recursion.

2. Look up Fractals on the Internet. Explain how these are examples of recursion.

3. Look up the Towers of Hanoi Puzzle. Play the game here: http://www.coolmath-games.com/0-tower-of-hanoi . How does this problem use recursion? ���� Consolidation:

HW. pg. 461 #1-4

Σ

The Binomial Theorem:

( ) knkn

k

n axk

nax −

=∑

=+

0

7. Sigma Notation, Pascal’s Triangle and the Binomial Theorem ���� Sigma Notation is a compact way of writing a lot of addition.

Example: nnn

i

ii xxxxxx 2...22222 44332211

1

+++++=∑=

Write the following in sigma notation:

3 + 4 + 5 + …

25 + 5 + 1 + … Watch the video on the Summation: https://www.youtube.com/watch?v=bAgCeNUvu0E (1:40 mins)

���� Pascal’s Triangle: Named after the famous mathematician Blaise Pascal

can you see how to find the next row of the triangle? Hint: It uses recursion.

Watch the video on the Pascal’s Triangle for the answer:

https://www.youtube.com/watch?v=Zo2JrPjijHc (1 min) So what does this have to do with Summation and Binomials? ���� Binomial Theorem: Take the time now to expand these fully. Write out your finally expansion: (x + y)0 = (x + y)1 = (x + y)2 = (x + y)3 = (x + y)4 = (x + y)5 = Now look at the numerical coefficients closely. Recognize them? What are the powers of x doing? What are the powers of y doing? Here is the Binomial Theorem: It writes what you just noticed in compact form.

Note:

k

n means “n choose k” and there is a button on your

calculator that lets you do the calculation so that you don’t have to write out all the rows of Pascal’s Triangle.

Examples: 1. Expand and simplify:

a) (x + y)8 = 807162534435261708 18285670562881 yxyxyxyxyxyxyxyxyx ++++++++ From the eighth row of the triangle and decreasing the powers of x and increasing the powers of y. b) (a – 2b)5 504132231405 )2(1)2(5)2(10)2(10)2(5)2(1 babababababa −+−+−+−+−+−=

5413223145 32808040101 bbabababaa −+−+−= From the fifth row of the triangle and decreasing the powers of a and increasing the powers of (-2b). ���� Your turn: 1. Expand and simplify:

a) (x + 3)4 = b) (1+ x2)3= c) 4

1

−x

x Hint: Write x

1 as 1−x

Watch the video for more examples: https://www.youtube.com/watch?v=bsFzOaVKVJg (10 mins)

���� Consolidation: 1. Write out the complete expansion of the following binomial expressions.

a) 5)( ba − b) 6)1( x+ c) 4)2( yx − d) 52)1( s− e) 7)1

(x

x +

2. Determine the general term for each of the following binomial expansions. a) 15)31( x− b) 20)105( x− c) 132)( xa − d) 252)5( xx − ���� Extension: 1. Determine the first four terms in the expansion 822 )1)(1( xxx −++ .

2. In the expansion of 78 )21()1( xx −−+ determine the coefficient of 5x .

3. In the expansion nxx 22 )1)(1( +− the third term is 2189x . Determine n.

4. In the expansion 8)( xa + the coefficient of 7x is 24. Determine a.

8. Assignment Sequences & Series Assignment