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• Students should know Arithmetic
sequence and series; sum of finite
arithmetic series; geometric sequences
and series; sum of finite geometric series
Arithmetic Sequence
Arithmetic Sequences
• An arithmetic sequence is a sequence in
which each term differs from the pervious
one by the same fixed number
• Example
– 2,5,8,11,14
• 5-2=8-5=11-8=14-11 etc
– 31,27,23,19
• 27-31=23-27=19-23 etc
Algebraic Definition
• {an} is arithmetic  an+1 – an= d for all
positive integers n where d is a constant
(the common difference)
–  “If and only if”
– {an} is arithmetic then an+1 – an is a constant
and if an+1 – an is constant the {an} is arithmetic
The General Formula
• a1 is the 1st term of an arithmetic sequence
and the common difference is d
• Then a2 = a1 + d therefore a3 = a1 + 2d
therefore a4 = a1 + 3d etc.
• Then an = a1 + (n-1)d
the coefficient of d is one less than the
subscript
No common difference!
Arithmetic
Sequence
Example #1
• Consider the sequence 2,9,16,23,30…
– Show that the sequence is arithmetic
– Find the formula for the general term Un
– Find the 100th term of the sequence
– Is 828, 2341 a member of the sequence?
Geometric Sequence
Arithmetic Sequences
1, 4, 7, 10, 13...
8,15,22,29...
•ADD To get next term
•Have a common difference
Geometric Sequences
2, 4, 8, 16, 32..
1
9,  3, 1,  ..
3
•MULTIPLY to get next term
•Have a common ratio
In a geometric sequence, the ratio of any term to the
previous term is constant.
You keep multiplying by the SAME number
each time to get the sequence.
This same number is called the common ratio
and is denoted by r
What is the difference between an arithmetic sequence
and a geometric sequence?
Try to think of some geometric sequences on your own!
8
16
32
 2,  2,  2
4
8
16
24
72
216
 3,
 3,
3
8
24
72
24
96
384
 4,
 4,
4
6
24
96
No common ratio!
10
15
 2,  1.5
5
10
Geometric
Sequence
To write a rule for the nth term of a geometric sequence,
use the formula:
nth term of geometric sequence an  a1r n1
r  common ratio
a1  First term
an  nth term
n  number of terms
Write a rule for the nth term of the sequence
6, 24, 96, 384, . . ..
Then find a
7
an  a1r
n1
a1  6
n 1
an  6(4)
24
r  6 4
To find a , plug 7 in for n.
7
n7
an  6(4) n 1
7 1
an 
an  6(4)
an  6(4)
6
This is the general
rule. It’s a
formula to use to
find any term of
this sequence.
an  6(4096)  24,576
Write a rule for the nth term of the sequence
1, 6, 36, 216, 1296, . . ..
Then find a
8
a1  1
6
r  1 6
n 8
an 
an  a1r
n1
an  1(6)
This is the general
rule. It’s a
formula to use to
find any term of
this sequence.
n 1
To find
a8 , plug 8 in for n.
n 1
an  1(6)
81
an  1(6)
7
an  1(6)
an  6  279,936
7
Write a rule for the nth term of the sequence
7, 14, 28, 56, 128, . . ..
Then find a6
One term of a geometric sequence is
r = 3. Write a rule for the nth term.
a1 
r 3
n3
an 18
a3  18
The common ratio is
One term of a geometric sequence is
a3  20 and one term is a6  160
Step 1: Find r
-divide BIG
160
8
20
small
-find the distance between the two terms
and take that root.
63  3
3
Step 2: Find a1 . Plug r, n, and an into your
equation. Then, solve for a1 .
Step 3: Write the equation using r and
a1 .
an  5(2) n 1
82
r2
an  a1r n 1
20  a1 231
20  a1 2 2
20  a1 4 5  a1
Write the rule when
a2  12
and
a4  192 .
an  3(4) n 1
Series and Sequences
Formulas
Let’s graph the sequence we just did.
Create a table of values. What kind of
function is this?
What is a? What is b?
Why do we pick all positive whole
numbers?
Domain, Input, X
Range, Output, Y
Compound Interest Formula
• P dollars invested at an annual rate r,
compounded n times per year, has a
value of F dollars after t years.
 r
F  P  1  
 n
nt
• Think of P as the present value, and F as
the future value of the deposit.
Compound Interest
• So if we invested $5000 that was
compounded quarterly, at the end of a
year looks like:
41
0.055 

F1  50001 
  $5280.72
4 

• After 10 years, we have:
0.055 

F10  50001 

4 

410
 $8633.85
Series and Sequences
Formulas