1. Chapter 1 Sequences and Series
1.1 Arithmetic Sequences
(Old Curriculum)

2. 1.1 Arithmetic Sequences
Many patterns and designs linked to mathematics are found in nature and the human body.
Certain patterns occur more often than others.
Logistic spirals, such as the Golden Spiral, are based on the Fibonacci number sequence.
The Fibonacci sequence is often called Nature’s Numbers, since this pattern is so commonly observed in nature. Often in the form of the Golden Spiral
Snail Shell
1, 1, 2, 3, 5, 8,
13…
Inner Ear
Fibonacci Sequence Video
Flower Petals
The Golden Spiral
Pre-Calculus 11

3. WHO AM I?
In 1705, Edmond Halley predicted that the comet seen in 1531, 1607, and 1682 would be seen again in Comets are made of frozen lumps of gas and rock and are often referred to as icy mud balls or dirty snowballs. Halley’s prediction was accurate. How did he know?
Edmond Halley
Halley’s Comet sightings in 1531, 1607, 1682, 1758 are approximately 76 years apart. They make a sequence.
→ sequence of the first 5 letters of the alphabet
{1, 2, 3, 4, …}
→ infinite sequence
{a, b, c, d, e}
{3, 5, 7, 9, …}
→ infinite sequence
{0, 1, 0, 1, 0, 1 …}
→ sequence of alternating 0’s and 1’s (yes they are in order, alternating order)
{3, 6, 9, 12}
→finite sequence
{4, 3, 2, 1}
→ backwards finite sequence
Pre-Calculus 11
Finding Patterns

4. What is an Sequence?
A sequence is an ordered list of numbers usually separated by commas.
An ordered list can mean anything we want. The order could be forward, backwards, …. alternating, or anytime of order we want!
The numbers in sequences are called terms.
t1 represents the 1st term, t2 - 2nd term, t3 - 3rd term ….
Pre-Calculus 11

5. Investigating Patterns
Sort the sequences into two groups.
What characteristic did you use to sort the lists?
1, 3, 5, 7, …
-7, -4, -1, 2, …
0, 5, 6, 12, …
-7, -6, -4, -1, …
2, 4, 8, 16, …
10, 20, 30, …
Arithmetic Sequences
NOT Arithmetic Sequences
1, 3, 5, 7, …
0, 5, 6, 12, …
-7, -4, -1, 2, …
2, 4, 8, 16, …
10, 20, 30, …
-7, -6, -4, -1, …
What are some possible characteristics of arithmetic sequences?
Pre-Calculus 11

6. What is an Arithmetic Sequence?
An arithmetic sequence is an ordered list of terms in which the difference between consecutive terms is a constant. The value added to each term to create the next term is the common difference.
common difference 14
2, 4, 6, 8, 10, 12, _____
7, 3, -1, -5, -9, _____ 2 -4
-13
Example
Given the sequence -5, -1, 3 …
a) What is the value of t1? -5
t3? 3
t4? 7
b) Determine the value of the common difference.
d = t2 - t1
= ( -1) - ( -5)
= 4
c) What strategies could you use to determine the value of t10?
Pre-Calculus 11

7. n tn Arithmetic Sequence Vocabulary
The terms of a sequence are labelled according to their position in the sequence.
The first term of the sequence is t1 (Sometimes referred to as term a)
The number of terms in the sequence can be represented by n.
The general term of the sequence (general rule) is tn. This term is dependent on the value of n.
Consider the sequence 3, 6, 9, 12, 15
The n value gives the relative position of each term. n tn
What assumptions are made?
read as “n is a natural number”
a natural number, N, is a positive integer (whole number), 1, 2, 3, etc.
3, 6, 9, 12, 15
The tn value gives the actual terms of the sequence.
Pre-Calculus 11

8. Deriving a Rule for the General Term of an Arithmetic Sequence
Consider the infinite sequence -5, -1, 3, 7, ...
Terms
Sequence
Sequence Expressed
using first term and
common difference
General Sequence -5 -1 3 7 -5
-5 + (4)
-5 + (4) + (4)
-5 + (4) + (4) + (4)
-5 + (4) +… + (4)
An arithmetic sequence is a sequence that has a constant
common difference, d, between successive terms.
tn = t1 + (n - 1)d
General
term or
nth term
Position of
term in
the sequence
First
term
Common
difference
Pre-Calculus 11

9. Consider the sequence -5, -1, 3 …
Example
Consider the sequence -5, -1, 3 …
Determine the value of t10.
b) Write the expression for the general term.
t1 = -5
n = 10
d = 4
t10 = ?
tn = t1 + (n - 1) d
- parameters t1 and d must be defined in the general term
t10 = -5 + (10 - 1) 4
= (9) 4
t10 = 31
t1 = -5
n = var
d = 4
tn = t1 + (n - 1) d
= -5 + (n - 1) 4
= n - 4
tn = 4n - 9
c) Use the general term to determine the value of t10
tn = 4n - 9
t10 = 4(10) - 9
t10 =
t10 = 31
Pre-Calculus 11

10. General Formula for an Arithmetic Sequence
An arithmetic sequence is a sequence that has a constant
common difference, d, between successive terms.
tn = t1 + (n - 1)d
General
term or
nth term
First
term
Position of
term in
the sequence
Common
difference
Which of the following sequences are arithmetic?
For the arithmetic sequences, what is the value of d?
a) 4, 7, 10, 13, …
b) 12, 7, 2, –3, …
c) 5, 15, 45, 135, …
d) x, x2, x3, x4, …
e) x, x + 2, x + 4, x + 6, …
d = 3
d = -5
d = 2
What is the value of t1?
t1 = x
Pre-Calculus 11

11. Determining tn tn = t1 + (n - 1)d tn = 5n + 65 tn = 70 + (n - 1)5
Many factors affect the growth of a child. Medical and health officials encourage parents to keep track of their child’s growth. The general guideline for the growth in height of a child between the ages of 3 years and 10 years is an average increase of 5 cm per year. Suppose a child was 70 cm tall at age 3.
a) Write the general term , in simplified form, that could be used to estimate what the child’s height will be at any age between 3 and 10.
t1 =
d = 70
tn = t1 + (n - 1)d
tn = 5n + 65
← general term 5
tn = 70 + (n - 1)5
What is the shape of this function?
tn = n - 5
f(n) = 5n + 65
general term notation only deals with a single variable “n” , this notation is more simplified
Name two differences between the notations.
b) How tall is the child expected to be at age 10?
tn = 5n + 65
What value of n would be used for age 10?
- n = 8 in this case because the child will still continue to grow for a year at age 10 before reaching age 11
t8 = 5(8) + 65
We would expect the child to be 105 cm tall.
t8 = 105
Pre-Calculus 11

12. Determining The Number of Terms
The musk-ox and the caribou of northern Canada are hoofed mammals that survived the Pleistocene Era, which ended years ago. In 1955, the Banks Island musk-ox population was approximately 9250 animals. Suppose that in subsequent years, the growth of the musk-ox population generated an arithmetic sequence, in which the number of musk-ox increased by approximately 1650 each year.
a) Write the first three terms of the arithmetic sequence
9250,
10 900,
12 550, …
b) How many years would it take for the musk-ox population to reach ?
Need to determine the value of n
tn = t1 + (n - 1)d
= (n - 1)1650
9250, , , …
= (n - 1)1650 t1 t2 t3 tn
55 = n - 1
56 = n
tn =
d = 1650
Since the value is the 56th term in the sequence, it would take 56 years for the musk-ox population to reach
The year would be 2011
Pre-Calculus 11

13. Calculating the first term and common difference
In an arithmetic sequence, the third term is 11 and the eighth term is 46. Determine the first term of the sequence and the common difference.
tn = t1 + (n - 1)d
solve linear system of equations for d
sub d into either equation to determine t1
t3 = 11
11 = t1 + 2d
46 = t1 + 7(7)
46 = t1 + 49
46 = t1 + 7d
-(11 = t1 + 2d)
Linear
system
t8 = 46
46 = t1 + 7d
35 = 5d
7 = d
t1 = -3
The first term is -3 and the common difference is 7.
Alternate method:
sub d into either equation to determine t1
t3 + d = t4
t3 + d + d = t5 so t8 = ….
t3 = t1 + (3 - 1)7
11 = t1 + 14
t3 , ___, ___, ___, ___, t8 t3
+ d
+ d
+ d
+ d
+ d
= t8
t3 + 5d = t8
t1 = -3
11 + 5d = 46
5d = 35
d = 7
Pre-Calculus 11

14. Name the common difference and first term given the general term for each arithmetic sequence.3 -2
First Term
= 7
= 16
Assignment
Suggested Questions
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1-11,13,16,19,21,26,27
Arithmetic Sequences
Pre-Calculus 11