Objective:
1. After completing activity 1, mod. 10
2. With 90% accuracy
3. -Identify sequences as arithmetic, geometric, or neither
 -Write recursive formulas for arithmetic and geometric
sequences
 -Write explicit formulas for arithmetic and geometric
sequences
 -Determine the number of terms in a finite arithmetic or
geometric sequence
Arithmetic Sequence:
An arithmetic (linear) sequence is a
sequence of numbers in which each new
term ( a n )is calculated by adding a
constant vale(d) to the previous term.
 For example: 1,2,3,4,5,6,…
The value of d is 1.
Find the constant value that is added to get
the following sequences & write out the
next 3 terms.
1. 2,6,10,14,18,22,…

Simple test to check if a pattern is
an arithmetic sequence:
Check that the difference between
consecutive terms is constant.
 For example, in the sequence:
1,2,3,4,5,6,… the constant is one because
6-5=5-4=4-3=3-2=2-1=1
In other words, since the difference is
constantly 1, then it is an arithmetic
sequence

Find the constant value that is added to get the
following sequences & write out the next 3 terms





2,6,10,14,18,22,… (you did this one
already)
-5,-3,-1,1,3,…
1,4,7,10,13,16,…
-1,10,21,32,43,54,…
3,0,3,-6,-9,-12,…
The recursive formula for an
arithmetic sequence
an  an 1  d
an  new term
an 1  previous term
d  constant
For example, the recursive formula for the arithmetic
sequence 1,2,3,4,5,6… is
an  an 1  1
Write the recursive formula for
each sequence:





2,6,10,14,18,22,…
-5,-3,-1,1,3,…
1,4,7,10,13,16,…
-1,10,21,32,43,54,…
3,0,3,-6,-9,-12,…
Answers:
an  an 1  4
an  an 1  2
an  an 1  3
an  an 1  11
an  an 1  3
Explicit formula

The explicit formula for a sequence
defines any term based on its term
number (n):
an  a1  d (n  1)
an  new term
a1  first term in pattern
d  costant
n  term number
Write the explicit formula for each
sequence





2,6,10,14,18,22,…
-5,-3,-1,1,3,…
1,4,7,10,13,16,…
-1,10,21,32,43,54,…
3,0,3,-6,-9,-12,…
Answers:
an  2  4( n  1)
an  5  2( n  1)
an  1  3( n  1)
an  1  11( n  1)
an  3  3( n  1)
Use your explicit formulas to
answer the questions: (show your
work)
1. What is the third term in the pattern:
2,6,10,14,18,22,…
2.What is the 20th term?
3.What is the 35th term?
Answers:
The 3rd term in the sequence
20th term in the sequence
an  2  4(n  1)
an  2  4( n  1)
an  2  4(3  1)
an  2  4( 20  1)
a n  2  4( 2)
an  2  4(19)
an 2  8
an  10
an  2 
an  78
35th term in the sequence
an  2  4(n  1)
an  2  4(35  1)
an  2  4(34)
a n  2  136
an  138
Geometric Sequence:
A geometric sequence- every term after
the first is formed by multiplying the
preceding term by a constant value called
the common ratio (or r)
 For example: 2,10,50,250,1250

◦ The value of r is 5 because :
1250
5
250
250
5
50
50
 5 and so on...
10
Simple test to check if a sequence is
a geometric sequence:
a3 a2
an

 r or
r
a2 a1
an 1
When you divide a term by a previous term you must arrive at
equal common ratios.
Determine the common factor for
the following geometric sequence:

5,10,20,40,80,…
1 1 1
,
, ,...
2 4 8
7,28,112,448,…
 2,6,18,54,…

Answer:
2
½
4
3

The recursive formula for a
geometric sequence
an  ra n 1
an  new term
an 1  previous term
r  common ratio
Write the recursive formula for each geometric sequence :
•5,10,20,40,80,…
• 1 , 1 , 1 ,...
2 4 8
•7,28,112,448,…
•2,6,18,54,…

•
•
•
•
Write the recursive formula
for each geometric sequence
5,10,20,40,80,…
1 1 1
, , ,...
2 4 8
7,28,112,448,…
2,6,18,54,…
Answers:
a n  2  a n 1
an
an
1

 a n 1
2
 4  a n 1
a n  3  a n 1
The explicit formula for a geometric
series is:
an  a1  r n 1
an  new term
a1  first term
r  common ratio
n  term number
Write the explicit formula for each geometric sequence :
•5,10,20,40,80,…
• 1 , 1 , 1 ,...
2 4 8
•7,28,112,448,…
•2,6,18,54,…
Answers:
a n  5  2 n 1
1 1
an   
2 2
an  7  4
an  23
n 1
n 1
n 1

Is the following sequence arithmetic or
geometric?: -3,30,-300,3000,….

Write a recursive & explicit formula for it.

Use the explicit formula to find the 8th
term.
Answer:



Geometric
an  10  an1
an  3  (10)
n 1
Warm-up 3/8/10

State whether the sequence is arithmetic,
geometric, or neither. Use your notes.

an  163  200 n

an  ( 2n)

2
an  4  ( 3)
n 1
Answers:
Arithmetic
 Neither
 Geometric

Warm-Up 3/9/10 (head your paper)
Consider the following arithmetic
sequences:
 0, 6, 12, 18,…120
 1, 9, 17, 25,…97
 15, 12, 9, 6,…-21
What is the common difference for each?
How many terms are in the sequence?

Class work:
Remember to head your notebook with
page # & today’s date.
 Old green Alg. II Book page 476 #1-9
under Exercises & Applications also do
#13 & 14 . Be Ready for a Quiz on it!
 Your regular book: Page 277 Assignment
#1.1, 1.3, 1.4, 1.9

Class work: 3/9/10
Old green Alg. II book page 479 #35-37
 Read the problem carefully


Your regular book Page 277 Assignment
#1.5 a-c
Homework:

Page 274-275 a-f.

Page 277 Warm-up #1-2
Homework Review, Check your
work:
Page 274 Discussion a.
 1. arithmetic
 2. geometric
 3. neither
 4. geometric
 5. neither
 6. Fibonacci

Discussion b.
1.) a 1  7; an  a n 1 6
1
2.) a 1  162; an  an 1  
3
Objective:
1.After completing activity 2, mod. 10
2. With 90% accuracy
3.-Identify sequences as arithmetic, geometric,
or neither
 Write explicit formulas for arithmetic and
geometric sequences
 Determine the number of terms in a finite
arithmetic sequence
 Write formulas for finite arithmetic series
Activity 2 Notes

Finite Series- the sum (S n)of the terms of
a finite sequence.
◦ For example: a finite series with n terms is:
Sn  a1  a2  a3  ...an
◦
Arithmetic Series: the sum of the terms of an
arithmetic sequence. For example:
Arithmetic Series : 2,4,6,8,10,12,14
S n  2  4  6  8  10  12  14  56
Find the sum of the first 100 natural
numbers:
Sn 
1  2  3  ... 98  99  100
 S n  100  99  98  ...3  2  1
------------------------------------------2S n  101  101  101  ...101  101  101
How many sets of 101 are there? 100
Therefore : 2S n  100 (101)
(divide both sides of equation by 2)
100 (101)
 5050
Sn 
2
The sum of the first 100 natural numbers is 5050.
Activity 2 Notes
Class work: page 281 exploration
 Parts a-c only

Answers to Exploration:
a1) Sn  500,500
n(1  n)
a 2) S n 
2
b) S n  2550
c ) S n  11,250
Conclusion Question:
What is the sum
of the following
finite series :
a1  a 2  a 3  ...a n 1  a n
Formula:
The sum of the terms of a finite
arithmetic sequence with n terms
and a common difference d can be found by :
n(a1  an )
Sn 
2
Where a1  first term of sequence
an  nth term
Formula 2:



The sum of the terms of a finite
arithmetic sequence with n terms & a
common difference d can also be found
by using the formula:
n
S n  2a1  (n  1)d
2

Please notice that this formula involves
the common difference d.
Warm-up: 3/11/10
Consider the sequence:
 7,11,15,…59
 Find the sum of all the terms


Answer: 462
Homework: 3/10/10
Warm –up page 282-283 #1-3
 Check your HW:
1. 2,001,000
2a. )3 2b.) 4 2c.) 113 2d. ) 25,561

3a.) 780
3b.) 1197
Class work: 3/11/10

Assignment page 283-284 #
2.1,
2.3,
2.4,
2.5,
2.6,
2.7
Answers to Assignment:
2.1) sum of first n even numbers: n(n+1)
 2.3) No because the sum of each pair is
not a constant
 2.4a.) the monthly payments can be
considered to be an arithmetic sequence
where the first term is $206.26 and the
common difference is $206.26
 2.4B) Yes. The payments form an
arithmetic sequence, their sum forms a
series.

Answers to Assignment:
2.4c) The lessee pays
$150  $1000  (36  $206 .26)  $8575 .36
of which $150 is refunded at the end of the lease.
2.4d) The difference of $2535.64 may be
the cost to purchase the car at the end of
the lease.
Answers to Assignment:
 2.5)
The number of newspapers delivered
each week forms an arithmetic sequencewhere
a1  15,000 and d  50. Answer : 846,300
2.6a) -2,3,8
 2.6b) 743
 2.6c) 55,575

Answers to Assignment:
2.7a) 2562.5
 2.7b) 2.875
 2.7c) 7.875 and 10.75

Objective
1.After completing activity 3, mod. 10
2.With 90% accuracy
3.Identify sequences as arithmetic, geometric, or
neither
 Write explicit formulas for arithmetic and
geometric sequences
 Determine the number of terms in a finite
geometric sequence
 Write formulas for finite geometric series
Geometric Series:
Geometric Series- the sum of the terms of
a geometric sequence.
For example:
2,6,18,54,162
The sum of those terms, in expanded form,
is the following geometric series :
S5  2  6  18  54  162  242
Explore:
Head your notebook with today’s date,
page # & title.
 With your partner, try the exploration on
page 284-285 parts a-g

Formula:


The sum of a finite geometric series with
n terms and a common ratio r:
a1r n  a1
Sn 
r 1
where a1 is the first term of the sequenceand r  1
where n is the stage number and r is the common ratio

Use the formula with the geometric
sequence: 2,6,18,54,162 to find the sum of
all 5 terms.

Warm-up page 286-287 #1-3
Assignment: # 3.1-3.4
 Skip part c for #3.3

Quiz Activity 2& 3 on Tuesday:
 Arithmetic & Geometric Series.You must
know Activity 1 to pass the quiz. 

warm-up
6
2
n

1

n2
Objective:
1.After completing activity 4, mod. 10
2. With 90% accuracy
3. Write explicit formulas for arithmetic and
geometric sequences
 Interpret the limit of an infinite sequence
 Determine the sum of the terms of an infinite
geometric sequence in which the common
ratio r is between –1 and 1
 Compare sequences that do and do not
approach limits