Sequences and Series
Mathematical Patterns
Suppose each student in your math class has a
phone conversation with every other member of
the class. What is the minimum number of calls
required?
Instead of actually making the calls, you can
represent telephone conversations by drawing
like the ones below.
?
How many calls are necessary for two people to have a conversation?
How many calls are necessary for everyone to talk to everyone else in a
group of three people? In a group of four people?
Use a diagram to find the number of calls needed for five people?
Which of the following expressions represent the pattern for number of
telephone calls?
2n  3
n(n 1)  5
How many calls would be needed for this class?
n(n 1)
2
Vocabulary
• Sequence – an ordered list of numbers that
can be described by a pattern.
• Term – any number in the sequence
Variables can be used to represent
terms of a sequence: a is typically
used.
1st term
2nd term
3rd term



a1
a2
a3
…
…
n - 1 term
nth term
n + 1 term



an-1
an
an+1
Recursive Formula
• Defines the terms of the sequence by relating
each term to the term before it.
{2, 4, 6, 8, 10…}
{0, 1, 3, 6, 10, 15}
Write a recursive formula for each sequence.
*Explicit formulas do not require the use or
knowledge of prior terms
Stacking Boxes
You are stacking boxes in levels that form
squares. The numbers of boxes in successive
levels form a sequence. The figure shows the
top five levels.
• How many boxes of equal size would you
need for the next lower level?
• How many boxes of equal size would you
need to add three levels?
Arithmetic Sequences
• The difference between consecutive terms in
the sequence is constant.
• This constant is called the COMMON
DIFFERENCE
6, 12, 18, 24 …
What is the common
difference?
Arithmetic Sequence Formulas
Recursive Formula
an  an 1  d
Explicit Formula
an  a1  (n 1)d
n is the term being described

d is the common difference
Arithmetic Mean
• Average between two values
a1  a2
2
Question: a2 of an arithmetic sequence
has a value of 6. a4 of the same
 has a value of 12. What is the
sequence
value of a3?
Geometric Series
• The ratio between consecutive terms is
constant.
Question: Why is it called a common ratio
(while in arithmetic sequences, it is called a
common difference)?
• Find the 6th term (or in this case, shape) in the
sequence.
• Write a formula that predicts how many
triangles will be present in the nth term.
Geometric Sequence Formulas
Recursive Formula
an  an 1 r
Explicit Formula
an  a1r
n 1
n is the term being described

r is the common ratio
Geometric Mean
• Geometric Mean between two positive values:
value 1value 2
Question: The second term of an geometric
sequence has a value of 8. The fourth
term of the same sequence has a value of
 18. What is the value of the third term?
The golfer
A particular golfer, sadly misses each putt on a
particularly bad hole. The ball ends up
halfway beyond the distance it started away
from the cup.
Draw a diagram showing
the result after each putt.
Write a sequence to represent the ball’s
distance from the hole after each putt if
the ball starts 12 feet from the hole.