Objective: 1. After completing activity 1, mod. 102. With 90% accuracy3. -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 ( )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,…

na

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

constant termprevious

termnew

1

1

daa

daa

n

n

nn

For example, the recursive formula for the arithmetic sequence 1,2,3,4,5,6… is

11 nn aa

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:

311324

1

1

1

1

1

nn

nn

nn

nn

nn

aaaaaaaaaa

Explicit formula The explicit formula for a

sequence defines any term based on its term number (n):

number termcostant

pattern in first term termnew

)1(

1

1

ndaa

ndaa

n

n

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:

)1(33)1(111

)1(31)1(25

)1(42

nana

nanana

n

n

n

n

n

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:

1082

)2(42)13(42)1(42

sequence in the term3rd The

n

n

n

n

n

aaaa

na

782

)19(42)120(42

)1(42sequence in the 20th term

n

n

n

n

n

aaaa

na

1381362

)34(42)135(42

)1(42 sequence in the 35th term

n

n

n

n

n

aaaa

na

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 :

on... so and 51050 5

50250 5

2501250

Simple test to check if a sequence is a geometric sequence:

r or 11

2

2

3 n

n

aar

aa

aa

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,…

7,28,112,448,…2,6,18,54,…

,...81,

41,

21

Answer:2½43

The recursive formula for a geometric sequence

ratiocommon r termprevious termnew

1

1

n

n

nn

aa

raa

Write the recursive formula for each geometric sequence :•5,10,20,40,80,…•

•7,28,112,448,…•2,6,18,54,…

,...81,

41,

21

Write the recursive formula for each geometric sequence

• 5,10,20,40,80,…•

• 7,28,112,448,…• 2,6,18,54,…

,...81,

41,

21

Answers:

1

1

1

1

3

421

2

nn

nn

nn

nn

aa

aa

aa

aa

The explicit formula for a geometric series is:

number termnratiocommon first term termnew

1

11

raa

raa

n

nn

Write the explicit formula for each geometric sequence :•5,10,20,40,80,…•

•7,28,112,448,…•2,6,18,54,…

,...81,

41,

21

Answers:

1n

1n

1

n

1n

32a

47a21

21a

25a

n

n

n

n

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

1

1

)10(3

10

n

n

nn

a

aa

Warm-up 3/8/10State whether the sequence is

arithmetic, geometric, or neither. Use your notes.

1

2

)3(4

)2(

200163

nn

n

n

a

na

na

Answers: ArithmeticNeitherGeometric

Warm-Up 3/9/10 (head your paper) Consider the following arithmetic

sequences: 0, 6, 12, 18,…1201, 9, 17, 25,…9715, 12, 9, 6,…-21What 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

#35-37Read 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. geometric5. neither 6. Fibonacci

Discussion b.

31 ;162a .)2

6 ;7a .)1

11

1n1

nn

n

aa

aa

Objective: 1.After completing activity 2, mod. 102. With 90% accuracy3.-Identify sequences as arithmetic,

geometric, or neither Write explicit formulas for arithmetic

and geometric sequences Determine the number of terms in a

finite arithmetic sequenceWrite formulas for finite arithmetic

series

Activity 2 NotesFinite Series- the sum ( )of the

terms of a finite sequence. ◦For example: a finite series with n

terms is: ◦

Arithmetic Series: the sum of the terms of an arithmetic sequence. For example:

nS

nn aaaaS ...321

561412108642,12,142,4,6,8,10 :Series ArithmeticnS

Find the sum of the first 100 natural numbers:

5050. is numbers natural 100first theof sum The

50502

)101(100S

2)by equation of sidesboth (divide )101(1002S :Therefore 100 there?are 101 of setsmany How

101101 ...101 101 101 101 2S -------------------------------------------

1 2 ...3 98 99 100S 1009998... 3 2 1

n

n

n

n

nS

Activity 2 Notes

Class work: page 281 explorationParts a-c only

Answers to Exploration:

250,11)2550)

2)1()2

500,500S )1 n

n

n

n

ScSb

nnSa

a

Conclusion Question:

n1n321 aa...aaa:series finite

following theof sum theisWhat

Formula:

nth term sequence of first term Where

2)(S

:by found becan d differencecommon a and n terms with sequence arithmetic

finite a of terms theof sum The

1

1n

n

n

aa

aan

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:

Please notice that this formula

involves the common difference d.

dnanSn )1(22 1

Warm-up: 3/11/10Consider the sequence: 7,11,15,…59Find the sum of all the terms

Answer: 462

Homework: 3/10/10Warm –up page 282-283 #1-3Check your HW: 1. 2,001,0002a. )3 2b.) 4 2c.) 113 2d. )

25,561

3a.) 780 3b.) 1197

Class work: 3/11/10Assignment 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 constant2.4a.) 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

2.4d) The difference of $2535.64 may be the cost to purchase the car at the end of the lease.

lease. theof end at the refunded is $150 which of36.8575$)26.206$36(1000$150$

Answers to Assignment: 2.5)

2.6a) -2,3,82.6b) 7432.6c) 55,575

846,300 :Answer 50.d and 000,15 wheresequence arithmetican formseach week

delivered newspapers ofnumber The

1 a

Answers to Assignment: 2.7a) 2562.52.7b) 2.8752.7c) 7.875 and 10.75

Objective1.After completing activity 3, mod. 102.With 90% accuracy3.Identify sequences as arithmetic,

geometric, or neitherWrite explicit formulas for arithmetic

and geometric sequencesDetermine the number of terms in a

finite geometric sequenceWrite 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

242162541862S:series geometric following theis

form, expandedin terms, thoseof sum The

5

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:

Use the formula with the geometric sequence: 2,6,18,54,162 to find the sum of all 5 terms.

ratiocommon theisr andnumber stage theisn where1r and sequence theof first term theis where

11

11

ar

araS

n

n

Warm-up page 286-287 #1-3

Assignment: # 3.1-3.4Skip 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

2 1n

n

Objective:1.After completing activity 4, mod. 102. With 90% accuracy3. Write explicit formulas for arithmetic

and geometric sequencesInterpret 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