Intent: Designed as an independent study unit focused on sequences and series, recursion
(Fibonacci Numbers, Koch’s snowflake, Sierpinski’s triangle) and Binomial Theorem.
Grade/Age: for MCR 3UI or MDM 4UI
Notes: This is designed to be something for students in grade 11 or 12 who need a little enrichment or in
some cases are missing some background.
Sequences, Series and the Binomial Theorem
Independent Study Unit
Due:_______________
ISU Outline
1. Sequences and Series – definitions and examples.
2. Two Special Types of Sequences – Arithmetic and Geometric.
3. Arithmetic and Geometric Series – Arithmetic and Geometric.
4. Recursion – Fibonacci Numbers, Koch’s snowflake, Sierpinski’s triangle
5. Binomial Theorem – Summation Notation, Sigma, Pascal’s Triangle
Instructions: The following is designed as an Independent Study Unit. You should try to read all the
notes and watch any indicated videos before you attempt the questions.
 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. Only final answers will be given.
 Extension:
These are questions or activities to extend your knowledge on the topic.
1. Sequences and Series
Definitions:
 A Sequence is a list (set) of numbers following a pattern in some kind of order.
Example:
a)
b)
c)
d)
1, 2, 3, 4, 5, …..
2, 4, 6, 8, …
2, 4, 8, 16, ….
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:
b) 2 + 4 + 6 + 8 +…
Formula: t n  n
Formula: tn  2n
c) 2 + 4 + 8 + 16 +….
Formula: t n  2 n
d) 0 + 1 + 4 + 9 + 16 + …
Formula: t n  n 2
where n starts at 1 and increases by 1 each time.
a) 1 + 2 + 3 + 4 + 5 +…..
 Your turn:
1. Given the formula for the nth term state the first 5 terms and any other given terms:
2

n  3
n 1
a) t n  5  2n , t50 ,t101
b) t n  2  1 , t10 , t15
c) t n 
, t12 ,t 25
n
2. Given the sequence find a formula for the nth term (if possible) and state the next 3 terms:
1 2 3 4
11 10 8 5
a) , , , ,...
b) 1,6,10,15,19,...
c) , , , ,...
2 3 4 5
3 3 3 3
 Consolidation:
HW. pg. 433 #1-4 , pg. 461 #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.
Example:
 Hint:
a) 2, 5, 8, 11, 14, …
b) 5, 3, 1, …
c) 18, 44, 70, …
1 2
d) , ,1,...
3 3
What is the common difference?
+3
+ (– 2) or – 2
+ 26
1
+
3
to find a “common difference” or “d” choose a term and subtract the previous term.
d  t2  t1 or in general d  t n  t n1
 Note: All sequences have an initial term or a first term, t1, 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
Given a, d, n
where a is the first term,
d is the common difference
and n is the term number
Formula: (General Term)
tn  a  (n  1)d
Example:
2, 5, 8, 11, 14, …
a = 2, d = 3, n = ?
Formula:
t n  2  (n  1)3
tn  2  3n  3
t n  1 3n
 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, …
9
ii) 5, ,4,...
2
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
2. Suppose two terms of an Arithmetic sequence are t3 = 11 and t9 = 29. Find the formula and the first 3
terms in the sequence.
3. Given two terms of a Arithmetic sequence find t n .
a) t5  16, t8  25
b) t4  2.5, t21  5
 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-sequencesintroduction (11 mins)
 A Geometric Sequence is a list (set) of numbers with a Common Ratio.
Example:
a) 2, 4, 8, …
b) 25, 5, 1, …
c) 3, -24, 192, …
1 2 4
d) , , ,...
3 3 3
 Hint:
What is the common ratio?
2
1

or  5
5
 (8)
2
to find a “common ratio” or “r” choose a term and divide it by the previous term.
t
t
d  2 or in general d  n
t n1
t1
 Note: All sequences have an initial term or a first term, t1, 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
Geometric
Given a, r, n
where a is the first term,
r is the common ratio
and n is the term number
Formula: (General Term)
 
t n  a r n1
Example:
3, -24, 192, …
a = 3, r = -8, n = ?
Formula:

t n  3 (8) n1

 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, 18, …
5 5
ii) 5, , ,...
2 4
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
5
ended at: i) 13122 ii)
128
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 t n .
a) t3  36, t 4  108
b) t3  99, t5  11
 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, …
2. State a, r or d and the formula for t n , then find the indicated term.
a) 2, 4, 8, …, t 7 and t12
b) 6, 0.6, 0.06, …, t8 and t15
c) 6, 8, 10, …, t10 and t 24
1 3 5
d) , , ,... t12 and t 21
2 2 2
e)
11 10 8 5
, , , ,...
3 3 3 3
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: tn  a  (n  1)d

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
2, 5, 8, 11, 14, …
Series
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: S n 
Example:
Given
5 + 9 + 13 +…
Then a = 5, d = 4, n = 12
n
2a  (n  1)d 
2
Find S12
12
2(5)  (12  1)( 4)
2
S12  610  (11)(4)
S12  610  44
S12  324
S12 
So the sum of the first 12 terms in the series is 324.
 Your turn:
1. Given the sequences:
9
i) 5   4  ... ii) 2 + 6 +18 + … iii) 2 + 6 + 10, … iv) – 3 + 6 – 12 + 24 …
2
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
v) 5 
5 5
  ...
2 4
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: t n  ar n1 

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
2x, 4x2, 8x3, …
Series
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: S n 
Example:
Given
5 + 20 + 80 +…
Then a = 5, r = 4, n = 7


a rn 1
r 1
Find S7


5 47  1
Sn 
4 1
516384  1
Sn 
3
S n  27305
So the sum of the first 7 terms in the series is 27305.
 Your turn:
1. Given the sequences:
9
5 5
i) 5   4  ... ii) 2 + 6 +18 + … iii) 2 + 6 + 10, … iv) – 3 + 6 – 12 + 24 …
v) 5    ...
2
2 4
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.
1
1
3. Suppose two terms of a Geometric Series are t1  and t 2  , find S6 and the formula for the Sn.
2
3
 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:
t n  t n2  t n1
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) t n  2t n1  n 2 , t1  2
b) t n  2t n2  t n1 , t1  1, t2  1
c) tn3  t n  2t n1  3tn2 ,
t1  12, t2  5, t3  1
2. Write an explicit formula for the nth term for the following:
1
1
1
a) t n  t n1  2 , t1  10
b) t n  t n1 , t1  20
c) t n  t n1 
, t1 
2
2
n(n  1)
 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?
7. Sigma Notation, Pascal’s Triangle and the Binomial Theorem
 Sigma Notation is a compact way of writing a lot of addition.
Σ
n
Example:
2 x
i
i
 21 x1  2 2 x 2  23 x 3  2 4 x 4  ...  2 n x n
i 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.
The Binomial Theorem:
x  a n   
n  k nk
x a
k 0  k 
n
n
Note:   means “n choose k” and there is a button on your
k 
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 = 1x8 y 0  8x 7 y1  28x 6 y 2  56 x 5 y 3  70 x 4 y 4  56 x 3 y 5  28x 2 y 6  8x1 y 7  1x 0 y 8
From the eighth row of the triangle and decreasing the powers of x and increasing the powers of y.
b) (a – 2b)5  1a 5 (2b) 0  5a 4 (2b)1  10a 3 (2b) 2  10a 2 (2b)3  5a1 (2b) 4  1a 0 (2b)5
 1a 5  10a 4b1  40a 3b 2  80a 2b 3  80a1b 4  32b 5
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:
4
a) (x + 3) =
1

c)  x  
x

2 3
b) (1+ x ) =
4
Hint: Write
1
as x 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) (a  b)5
b) (1  x) 6
c) ( x  2 y ) 4
d) (1  s 2 )5
2. Determine the general term for each of the following binomial expansions.
a) (1  3x)15
b) (5  10 x) 20
c) (a  x 2 )13
d) (5 x  x 2 ) 25
 Extension:
1. Determine the first four terms in the expansion (1  x  x 2 )(1  x 2 )8 .
2. In the expansion of (1  x)8  (1  2 x) 7 determine the coefficient of x 5 .
3. In the expansion (1  x 2 )(1  x) 2 n the third term is 189x 2 . Determine n.
4. In the expansion (a  x)8 the coefficient of x 7 is 24. Determine a.
1
e) ( x  ) 7
x