11.1
Series and Sequences
TermsSequence- a list of ordered numbers
Each individual number can be named a “term”
Ex:
Finite sequence vs infinite sequence
Rule-can be written as a function or in sequence notation
Ex.
Ex.1: Write the first 6 terms:
a)
an  5  n
b) f (n)  5n
Ex. 2: Describe the pattern by writing the rule. Then write the next term.
a)
2 2 2
2
, ,
,
5 25 125 625
b) 3,5,7,9
Graphing the sequence: (x,y) (Domain, Range)
Domain-position of each term
Range-individual term itself
Ex. 3: One year before you go on a trip you begin making deposits to an account.
The first month you deposit $40. For the next 11 months you deposit $3 more than
the previous month. Write a rule for the amount of each monthly deposit. Then
graph the sequence.
Series-the sum of the sequence
Ex. Sequence: 2,5,8,11,14
Series: 2 + 5+ 8+ 11+ 14
Summation Notation (Sigma Notation)-
#
 Rule
6
 3i  4
i#
i 1
I = index of notation
# after I = lower limit
# above = upper limit
Ex. 4: Write each series with summation notation
a) 4 + 8+12+…..+100
b) 2 +
3 4 5
   ....
4 9 16
Ex. 5: Find the sum of the series
10
k
2
1
k 5
SPECIAL FORMULAS for SPECIAL SERIES (consecutive integers)
n
n
1
n
i
i
i 1
i1
i 1
Shortcuts:
Ex.6:
5
1
:n
10
x
x 1
2
8
i
i 1
i1
Ex.8:
Ex. 7:
;
n(n  1)(2n  1)
6
:
n( n  1)
2
2
11.2
Arithmetic Series and Sequences
Arithmetic Sequences- a constant difference between terms “d”
Ex. 1: Decide if these are arithmetic sequences. Explain
a) -10, -6, -2, 0, 2, 6, 10
b) 5,11,17, 23, 29
Rule for any arithmetic sequence:
an = a + d(n -1)
1
Ex. 2: Write a rule for the nth term: 32, 47, 62, 77,……. Then find
a15 .
Ex. 3: One term is a20  111. The common difference is –6. Write the rule for
the nth term.
Ex. 4: Two terms of an arithmetic sequence are a5  10 and a30  110 . Write the
rule for the nth term. Then find the value of “n” such that an  2 .
Arithmetic Series- the sum of an arithmetic sequence. Labeled S n
Rule for an arithmetic Series:
 a  an 

Sn  n  1


2


Ex. 5: Consider the arithmetic series: 20 + 18 +16 +14+…
a) Find sum of the first 25 terms:
b) Find n such that Sn  760
11.3 Geometric Series and Sequences
Ratio- constant quotient/ratio of 2 consecutive terms, which we call “r” found by:
Ex. 1: Is this an example of a geometric sequence?
a) 4, -8, 16, -32,….
b) 3, 9, -27, -81, 243
Rule for Writing a Geometric Sequence:
an  a r n1
1
Ex. 2: Write the rule for the nth term of 5,2,0.8, 0.32… Then find a8 .
Ex. 3: One term of a geometric sequence is a4  3. The common ratio is r=3.
Write a rule for the sequence and then graph the sequence.
Ex. 4: Two terms of a geometric sequence are a2  4 and a6  1024 . Write a rule
for the nth term.
Sum of a Geometric Series-
 1 rn 
Sn  a1 

 1 r 
Ex. 5: Find the sum of the first 10 terms of the geometric series: 4 + 2 + 1 + ½ +….
Then find n such that S n 
31
.
4
Ex. 6: You buy a new car for $25,000. The value decreases by 16% each year.
Write a rule for the average yearly value of the car. About how many years will the
value fall to $10,455 ?
11.4 Infinite Geometric Series
A finite series has a sum
An infinite series does and doesn’t have a sum.
If the ratio is between –1 and 1, you can still find the sum of an infinite geometric
series.
Consider the geometric series: ½+ ¼ + 1/8 + 1/16 + ……
It has infinitely many terms, but it does have a finite sum.
S1 
S4 
S7 
S2 
S5 
S8 
S3 
S6 
S9 
As n gets bigger, the sum gets closer and closer to what number?
As n gets bigger, ( ½ ) gets closer and closer to what number?
 1 rn 
Sn  a1 

 1 r 
Sum of an infinite geometric series with : 1  r  1
S
a1
1 r
Ex. 1: Find the sum of the infinite geometric series.
a)

 2(0.1)
i 1
i 1
b) 12+ 4 +
4 4
  .....
3 9
Ex. 2: An infinite geometric series with a1  5 has a sum of
27
. What is the
5
common ratio of the series?
11.5
Recursive Rules for Sequences
Explicit Rule- gives a term relative to the rule. You don’t necessarily need the
previous term.
Recursive Rule- gives the beginning term and then a recursive equation that shows
how each term is found by using the previous term.
Ex.1: Write the first five terms of the sequence:
a)
a1  2
a1  1
an   an 1   1
2
b) a2  2
an  an  2  an 1
Factorial-product of all integers from 1 to n.
Ex. 2: Write the rule for the arithmetic sequence where a1  15 and d=5.
a) explicit rule
b) recursive rule
Ex.3: Write the rule for the geometric sequence where a1  4 and r = 0.2.
a) explicit rule
b) recursive rule
Writing a recursive rule that is neither arithmetic nor geometric.
 Determine which term the pattern begins
 Give the terms of which the pattern doesn’t work for
 Give the pattern as a rule
Ex. 4: Write a recursive rule for the sequence: 1,1,4,10,28, 76.
Ex. 5: Write a recursive rule for the sequence: 2, 3, 8, 63, 3968, 15745023
Ex. 6: A nursery initially had 500 trees. Each year it sells 70% of its stock and
adds 500 new trees. Write a recursive rule for the number of trees at the
beginning of the nth year. How many trees does it have at the beginning of the
fifth year? What happens to the number of trees over time?