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Algebra II
Lesson 2: F 502.
Unit 1 Lesson 2, 3 & 5
Find the next term in a sequence described
recursively.
Lesson 3: F 603.
Find a recursive expression for the general term in a
sequence described recursively.
Lesson 5: Exhibit knowledge of geometric sequences.
Lesson 4: Solve complex arithmetic problems involving percent of increase /decrease or
requiring integration of several concepts
(i.e. using several ratios, comparing percentages or averages).
Do Now:
Identify each sequence as arithmetic, geometric or
neither.
c.
14, 7, 3.5, 1.75,…
47, 41, 35, 29,…
1, 1, 2, 3, 5, 8,…
d.
The non-horizontal cards
a.
b.
According to the US EPA, each American
produced an average of 978 lb of trash in 1960.
This increased to
1336 lb in 1980. By
2000, trash production
had risen to 1646 lb/yr
per person. How
would you find the
total amount of trash a
person produced in a
lifetime?
Arithmetic Series
Key Concepts:
We are learning about mathematical series
and the summation notation used to
represent them.
Write recursive and explicit formulas for the
terms of the series and find partial sums of
the series.
Vocabulary
Summation Symbol
When we don’t want to write out a whole bunch of numbers in the series, the
summation symbol is used when writing a series. The limits are the greatest
and least values of n.
Upper Limit (greatest value of n)
Summation symbol
Explicit function for
the sequence
Lower Limit (least value of n)
So, the way this works is plug in n=1 to the equation and continue through n=3.
(5*1 + 1) + (5*2 +1) + (5*3 + 1) = 33
Understanding Sigma Notation
Evaluate the following expressions
4
 3n
n 1
5
 (k
k 1
2
 2)
1
 (4 p  1)
p  2
What did Gauss do?
Examples
Ex. 1.) Find the sum of the first 50 multiples of 6:
6,12,18,...u50 
Ex.2.) Find the sum of the first 75 even numbers starting with 2.
Find the indicated values
18
1.
 6k  1
k 4
2.
S 9 if
un  5n  6
3.
u18 if
un  3n  4
The concert hall in the picture below has 59 seats in Row
1, 63 seats in Row 2, 67 seats in Row 3, and so on.
How many seats are there in all 35 rows of the concert
hall?
Suppose that Seat 1 is at the left end of Row 1 and that
Seat 60 is at the left end of Row 2. Describe the location
of Seat 970.
Practice
Page: 633 #(1 – 5)
Sum of a Finite Arithmetic Series
n
Sn  ( a1  an )
2
Let’s try one: evaluate the series:
5, 9, 13,17,21,25,29
7
Sn  (5  29)  119
2
Writing the given series in summation form.
Evaluate: Yes, you can add manually. But let’s try
using the shortcut:
n
Sn  ( a1  an )
2
Sn 
6
(102  112)  642
2
Practice
Find the number of terms, the first term and the last term.
Then evaluate the series:
Ex.1
Ex.2
Why is the
answer not 58?
Notice we can use the
shortcut here:
Sn 
10
( 2  7)  5(5)  25
2
Note: this is
NOT an
arithmetic
series. You can
NOT use the
shortcut; you
have to
manually
calculate all
values.
Arithmetic Sequence:
The difference between consecutive
terms is constant (or the same).
The constant difference is also known as
the common difference (d).
(It’s also the number you are adding
every time!)
The general form of an ARITHMETIC sequence.
First Term:
a1
Second Term:
a2  a1  d
Third Term:
a3  a1  2d
Fourth Term:
a4  a1  3d
Fifth Term:
a5  a1  4d
nth Term:
an  a1   n 1 d
Formula for the nth term of an ARITHMETIC
sequence.
an  a1   n 1 d
an  The nth term
a1  The 1st term
n  The term number
d  The common difference
Example: Decide whether each
sequence is arithmetic.
-10,-6,-2,0,2,6,10,…
-6-(-10)=4
-2-(-6)=4
0-(-2)=2
2-0=2
6-2=4
10-6=4
Not arithmetic (because
the differences are not
the same)
5,11,17,23,29,…
11-5=6
17-11=6
23-17=6
29-23=6
Arithmetic (common
difference is 6)
Rule for an Arithmetic
Sequence
n = number of terms
an = last term
an= a1+(n-1)d
Example: Write a rule for the nth
term of the sequence 32,47,62,77,… .
Then, find u12.
There is a common difference where d=15,
therefore the sequence is arithmetic.
Use un=u1+u(n-1)d
un=32+(n-1)(15)
un=32+15n-15
un=17+15n
u12=17+15(12)=197
Example: One term of an arithmetic sequence
is a8=50. The common difference is 0.25.
Write a rule for the nth term.
Use an=a1+(n-1)d to find the 1st term!
a8=a1+(8-1)(.25)
50=a1+(7)(.25)
50=a1+1.75
48.25=a1
* Now, use an=a1+(n-1)d to find the rule.
an=48.25+(n-1)(.25)
an=48.25+.25n-.25
an=48+.25n
Example: Two terms of an arithmetic sequence are
a5=10 and a30=110. Write a rule for the nth term.
Begin by writing 2 equations; one for each term
given.
a5=a1+(5-1)d OR 10=a1+4d
And
a30=a1+(30-1)d OR 110=a1+29d
Now use the 2 equations to solve for a1 & d.
10=a1+4d
110=a1+29d (subtract the equations to cancel a1)
-100= -25d
So, d=4 and a1=-6 (now find the rule)
an=a1+(n-1)d
an=-6+(n-1)(4) OR an=-10+4n
Arithmetic Series
The sum of the terms
in an arithmetic
sequence
The formula to find
the sum of a finite
arithmetic series is:
1st Term
Last
Term
 a1  an 
S n  n

 2 
# of terms
Example: Consider the arithmetic
series 20+18+16+14+… .
Find n such that Sn=-760
 a1  an 
S n  n

 2 
 20  (22  2n) 
 760  n

2


 a1  an 
S n  n

 2 
 20  28 
S 25  25
 S 25  25(4)  100
2


 20  (22  2n) 
 760  n

2


-1520=n(20+22-2n)
-1520=-2n2+42n
2n2-42n-1520=0
n2-21n-760=0
(n-40)(n+19)=0
n=40 or n=-19
Always choose the positive solution!
An introduction…………
Sequence
Sum
Sequence
Sum
1, 4, 7, 10, 13
35
2, 4, 8, 16, 32
62
9, 1,  7,  15
12
9,  3, 1,  1/ 3
20 / 3
6.2, 6.6, 7, 7.4
27.2
,   3,   6
3  9
1, 1/ 4, 1/16, 1/ 64 85 / 64
9.75
, 2.5, 6.25
Arithmetic Sequences
Geometric Sequences
ADD
To get next term
MULTIPLY
To get next term
Vocabulary of Sequences (Universal)
a1  First term
an  nth term
n  number of terms
Sn  sum of n terms
d  common difference
nth term of arithmetic sequence  an  a1  n  1 d
sum of n terms of arithmetic sequence  Sn 
n
 a1  an 
2
The sum of the first n terms of an infinite
sequence is called the nth partial sum.
Sn  n (a1  an)
2
Vocabulary of Sequences (Universal)
a1  First term
an  nth term
n  number of terms
Sn  sum of n terms
r  common ratio
nth term of geometric sequence  an  a1r n1


a1 r n  1 

sum of n terms of geometric sequence  Sn  
r 1
1, 4, 7, 10, 13, ….
Infinite Arithmetic
3, 7, 11, …, 51
Finite Arithmetic
1, 2, 4, …, 64
Finite Geometric
1, 2, 4, 8, …
1 1 1
3,1, , , ...
3 9 27
No Sum
n
Sn   a1  an 
2
a1 r n  1
Sn 
r 1
Infinite Geometric
r>1
r < -1
No Sum
Infinite Geometric
-1 < r < 1
a1
S
1 r
Find the sum, if possible:
1 1 1
1     ...
2 4 8
1 1
1
2
4
r  
 1  r  1  Yes
1 1 2
2
a1
S

1 r
1
1
1
2
2
Find the sum, if possible:
2 2  8  16 2  ...
8
16 2
r

 2 2  1  r  1  No
8
2 2
NO SUM
Find the sum, if possible:
2 1 1 1
  
 ...
3 3 6 12
1 1
1
3
6
r  
 1  r  1  Yes
2 1 2
3 3
a1
S

1 r
2
3
4

1 3
1
2
Find the sum, if possible:
2 4 8
   ...
7 7 7
4 8
r  7  7  2  1  r  1  No
2 4
7 7
NO SUM
Find the sum, if possible:
5
10  5   ...
2
5
5
1
2
r
 
 1  r  1  Yes
10 5 2
a1
10
S

 20
1
1 r
1
2
The Bouncing Ball Problem – Version A
A ball is dropped from a height of 50 feet. It rebounds 4/5 of
it’s height, and continues this pattern until it stops. How far
does the ball travel?
50
40
40
32
32
32/5
32/5
S
50
40

 450
4
4
1
1
5
5
The Bouncing Ball Problem – Version B
A ball is thrown 100 feet into the air. It rebounds 3/4 of
it’s height, and continues this pattern until it stops. How far
does the ball travel?
100
100
75
75
225/4
225/4
S
100
100

 800
3
3
1
1
4
4
Sigma Notation
UPPER BOUND
(NUMBER)
B
SIGMA
(SUM OF TERMS)
a
n A
n
NTH TERM
(SEQUENCE)
LOWER BOUND
(NUMBER)
4
  j  2  1  2   2  2   3  2    4  2  18
j1
7
  2a    2  4    2 5    2  6   2 7  44
a4

4
n 0
 
 
 
 
 
4
3
2

0.5

2

0.5

2

0.5

2
0.5  2  0.5  2  0.5  2
n
 33.5
0
1


n
0
2
1
3
3
3
3
6

6

6

5
 5   6 
 5   ...

 
 
b 0 
5
a1
S

1 r
6
3
1
5
 15
  2x  1   2 7  1   2 8  1   2 9  1  ...   2  23  1
23
x 7
n
23  7  1
Sn   a1  an  
15  47   527
2
2
  4b  3    4  4  3   4 5  3   4  6  3   ...   4 19  3
19
b 4
Sn 
n
19  4  1
a

a

 1 n
19  79   784
2
2
Rewrite using sigma notation: 3 + 6 + 9 + 12
Arithmetic, d= 3
an  a1  n  1 d
an  3  n  1 3
an  3n
4
 3n
n1
Rewrite using sigma notation: 16 + 8 + 4 + 2 + 1
Geometric, r = ½
an  a1r n1
n1
 1
an  16  
2
 1
16  

2
n1
5
n1
Rewrite using sigma notation: 19 + 18 + 16 + 12 + 4
Not Arithmetic, Not Geometric
19 + 18 + 16 + 12 + 4
-1
-2 -4 -8
an  20  2n1
5
n1
20

2

n1
3 9 27


 ...
5 10 15
Rewrite the following using sigma notation:
Numerator is geometric, r = 3
Denominator is arithmetic d= 5
NUMERATOR:
DENOMINATOR:
3  9  27  ...  an  3 3 
n1
5  10  15  ...  an  5  n  1 5  an  5n

SIGMA NOTATION:

n1
3 3 
n1
5n