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Evaluate each expression for x = 4, 5, 6.
1. 2x + 1
9, 11, 13
2. 0.5x + 1.5
3.5, 4, 4.5
3. x2 - 1
4. 2x + 3
15, 24, 35
19, 35, 67
Section 9.4
WHY?
Infinite sequences, especially those with finite limits are involved in some
key concepts in calculus.
A sequence is an ordered set of numbers. Each
number in the sequence is a term of the
sequence. A sequence may be an infinite
sequence that continues without end, such as the
natural numbers, or a finite sequence that has a
limited number of terms, such as {1, 2, 3, 4}.
Instead of function notation, such as f(x) or a(n),
sequence values are written by using subscripts.
The first term is a1, the second term is a2, and the
nth or kth term is an or ak. Because a sequence is
a function, each number n or k has only one term
value associated with it, an or ak.
*** some books use n, some use k ***
Reading Math
an is read “a sub n.”

Example 1: Find the first 6 terms and the 100th
term of the sequence {ak}, in which ak = k2 – 1
akak
a1a1
Process:
Process:
k2k–2 1– 1
12 – 1
Term
Term
=0
a2a2
a3a3
a4a4
a5a5
22 – 1
32 – 1
42 – 1
52 – 1
=3
=8
= 15
= 24
a6a6
62 – 1
= 35
a100
a100
1002 – 1
= 9999
Go back to your warm up #3, does this match up for x = 4, 5, 6???

You can use your calculator by using the
function: y= x2 – 1
 Go to table set and choose ASK for Independent
 Then simply type the x values and press enter

You can also use your calculator by
typing the following commands:
0k
k+1k: k2 – 1
▪  is the STO> button
▪ k is the ALPHA character for the ( button
▪ : is the ALPHA character for the . Button
Every time you press enter it will show you the next term starting with a1
Example 2: Find the first 6 terms and the 100th term of the
sequence defined recursively by the conditions:
b1 = 3
bn = bn-1 + 2
bn
b1
b2
b3
b4
b5
b6
for all n > 1
Process
b2-1 + 2  b1 + 2  3+2
Term
=3
=5
=7
=9
= 11
= 15
Example 2: Find the first 6 terms and the 100th term of the
sequence defined recursively by the conditions:
b1 = 3,
bn = bn-1 + 2
for all n > 1
Can you see that pattern or sequence?
{3, 5, 7, 9, 11, 13, …}
To get the 7th term, it is 6 terms beyond the 1st which
means that we can quickly get there by adding 6 2’s to
the 1st term, 3.
b7 = 3 + 6(2) = 15 Is this reasonable for the sequence?
Now find the 100th term:
201

You have probably heard of the Fibonacci
Sequence. One way to propagate the
sequence is with the help of Pascal’s Triangle:
=1
=1
=2
1 =3
=5
2 1 =8
1
1
1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
You can find each term of the
sequence using addition,
but the sequence is not
arithmetic. The recursive
formula for the Fibonacci
sequence is:
Fn = fn-2 +Fn-1
F1 =1
F2 = 1
Graphs of Arithmetic Sequences
relate to linear functions
(think adding for consecutive terms)

Graphs of Geometric Sequences
relate to exponential functions
(think multiply for consecutive terms)

Explicit:
Explicit:
Recursive:
a1 = a given value
an = an-1 + d
(for all n > 2)
Recursive:
a1 = a given value
(for all n > 2)

Example 3: Identify the given sequences as
arithmetic, geometric, neither or either?
Start by checking common differences and common ratios
 -5, 10, -20, 40, …
Geometric, the common ratio is -2
 2, 2, 2, 2, …
Either, the common difference is 0
And the common ratio is 1
 25, 50, 75, 100, …
Arithmetic, the common difference is 25
 1, 4, 9, 16, …
Neither, there are no common difference
or common ratio

The Arithmetic mean is simply adding 2
numbers and dividing by 2, it is what you
think of when you think “average”

Example 4: Find the missing term of the
arithmetic sequence 84, ___, 110, …

Example 5: Find the missing term of the
arithmetic sequence 24, ___, 57, …

Example 6: Find the missing term of the
geometric sequence 3, ___, 18.75, …

Example 7: Find the missing term of the
geometric sequence 9180 , ___, 255, …

Example 8: Write a recursive and an explicit
formula for the given arithmetic sequence:
-32, -20, -8, 4, 16
d, (common difference) = 12,
Explicit:
an = a1 + (n-1)d
an = -32 + (n-1)12
a1, (1st term) = -32
Recursive:
a1 = the first term
an = an-1 + d (for all n > 2)
a1 = -32
an = an-1 + 12, (for all n > 2)

Example 9: Write a recursive and an explicit
formula for the given geometric sequence:
3, 6, 12, 24, 48, …
r, (common ratio) = 2,
Explicit:
an = a1 * r n-1
an = (3) 2 n-1
a1, (1st term) = 3
Recursive:
a1 = the first term
an = an-1 * r (for all n > 2)
a1 = 3
an = (2)an-1
(for all n > 2)

Just as we are concerned with the end behavior
of functions, we will also be concerned with the
end behavior of sequences.
Converges
If the degree of the
numerator is the same as
the degree of the
denominator, the limit is
the ratio of the leading
coefficients
If the degree of the
numerator is less than the
degree of eh
denominator, the limit is
zero.
Diverges
If the degree of the
numerator is greater than
the degree of the
denominator, the limit is
infinite.

A sequence is a pattern

A series is the sum of the terms in the
sequence
Finite Sequence
6, 9, 12, 15, 18
Finite Series
6 +9 + 12 + 15 + 18
Infinite Sequence
3, 7, 11, 15, …
Infinite Series
3 + 7 + 11 + 15 + …

For an infinite Geometric series, use the
following, where a1 is the 1st term and r is the
common ratio.

Just as we are concerned with the end
behavior of functions, we will also be
concerned with the end behavior of series.
Converges
Diverges, (no limit)
|r|<1
|r|>1
*converges to the sum S
*diverges, no limit
 9.4 p.739, 1-31 odd
 Project due 5/2 or 5/3
 Quiz at the end of next class