U NIT 7 Sequences & Series. P ATTERNS E XAMPLE : o Suppose every student in class has a phone...

UNIT 7Sequences & Series

PATTERNS

EXAMPLE:o Suppose every student in class has a phone

conversation with every other member of the class.

o What is the minimum number of calls required?o We can show telephone conversations by using

the following diagrams:

?

CONTINUED….o How many calls are necessary for two people

to have a conversation?o How many calls are necessary for everyone to

talk to everyone else in a group of three people?

o In a group of four people?o Use a diagram to find the number of calls

needed for five people.

1 call

3 calls

6 calls

10 calls

CONTINUED…o Which of the following formulas can you use to

find the pattern for the telephone calls?

o Use the formula from the previous question to find the number of calls needed for a group of 7 students.

o How many calls would be needed for this class?

a.) 2 3n b.) 1 5n n c.) 1

2

n n C

1

2

n n 7 7 1

2

7 6

2 42

2 21

Some patterns can be described with a sequence, or a list of numbers.

Each number in a sequence is a term.

EXAMPLE: Describe the pattern formed & find the next

three terms.

a.) 27, 34, 41, 48 Add 7; 55, 62, 69

b.) 243, 81, 27, 9 Divide by 3; 3,1,1/3

EXAMPLE: Suppose you drop a handball from a height of 10 ft. After the ball hits the floor, it rebounds to 85% of its

previous height.

a.) How high will the ball rebound after its fourth bounce?

Original height of the ball: 10 ft.

After 1st bounce:

After 2nd bounce:

After 3rd bounce:

After 4th bounce:

The ball will rebound about 5.2 feet after the 4th bounce.

85% of 10 = .85 10 8.5 ft.85% of 8.5 = .85 8.5 7.225

85% of 7.225 = .85 7.225 6.14185% of 6.141 = .85 6.141 5.220

CONTINUED…b.) About how high will the ball rebound after the 7th

bounce?After 4th bounce:After 5th bounce:After 6th bounce:After 7th bounce:

The ball will rebound about 3.2 feet after the 7th bounce.

5.22085% of 5.220 .85 5.220 4.43785% of 4.437 .85 4.437 3.7714585% of 3.77145 .85 3.77145 3.206

CONTINUED… c.) After what bounce will the rebound height be less

than 2 feet?After 7th bounce:After 8th bounce:After 9th bounce:After 10th bounce:

After the 10th bounce, the rebound height will be less than 2 feet.

3.20685% of 3.206 .85 3.206 2.725185% of 2.7251 .85 2.7251 2.316385% of 2.3163 .85 2.3163 1.969

HOMEWORK

Worksheet: Patterns

RECURSIVE FORMULA

TERMS OF A SEQUENCE

A variable, such as a, with positive integer subscripts, can be used to represent the terms

of a sequence.

1st Term 2nd Term n-1 Term nth Term n+1 Term

a1 a2 an-1 an an+1

(previous term) (next term)

RECURSIVE FORMULA

A recursive formula defines the terms in a sequence by relating each term to the ones

before it.

The pattern in yesterday’s bouncing ball example was recursive because the height of

the ball after each bounce was 85% of its previous height.

The recursive formula that describes the ball’s height is , where a1=10

10.85n na a

(the first term in the sequence always needs to be defined for a recursive sequence)

EXAMPLE:a.) Describe the pattern that allows you to find

the next term in the sequence 2, 4, 6, 8, 10…. Write a recursive formula for the sequence.• Add 2 to a term to find the next term• A recursive formula is , where a1= 2

b.) Find the sixth & seventh terms in the sequence.• Since a5 = 10, a6 = 10+2 = 12

• Since a6 = 12, a7 = 12+2 = 14

1 2n na a

CONTINUED…c.) Find the value of term a9 in the sequence.

• Term a9 is the ninth term

• a9 =

d.) Find terms a11 & a15

• a11 =

• a15 =

8 2a 7 2 2a 14 2 2 18

10 2a 9 2 2a 18 2 2 22

14 2a 13 2 2a 12 2 2 2a

11 2 2 2 2a

22 2 2 2 2 30

EXAMPLE:Write a recursive formula for each sequence.

Then find the next term.

a.)

b.)

1 1 1 1 1, , , , ,...

2 4 8 16 32 1

1

2n na a 1

1,

2a

Next Term:Recursive Formula:1

64

9144, 36, 9, ,...

4

Recursive Formula:

1

4n

n

aa 1, 144a

Next Term:

9

16

HOMEWORK

Worksheet: Recursive Formulas

EXPLICIT FORMULAS

EXPLICIT FORMULAA recursive formula allows us to find the value of a

term by using the preceding term.

However, we can sometimes find the value of a term without knowing the preceding term by using the

number of the term to calculate its value.

An explicit formula expresses the nth term in terms of n.

EXAMPLE: The spreadsheet shows the

perimeters of squares with sides

from 1 to 6 units long. The numbers in each row form a sequence.

a.) For each sequence, find the next term (a7) & the twenty-fifth term (a25). Row 2: each term is the same as its subscript; therefore, a7

= 7 and a25 = 25 Row 3: each term is 4 times its subscript; therefore, a7 = 4(7) = 28 & a25 = 4(25) = 100.

CONTINUED…b.) Write an explicit formulafor each sequence.

Row 2: Row 3:

c.) Write the first six terms in the sequence showing the areas of the squares. Then find a20

Area:

d.) Write an explicit formula for the sequence from part (c).

na n4na n

1, 9,4, 16,25, 36; 400

2na n

HOMEWORK

Worksheet: Explicit Formulas

ARITHMETIC SEQUENCES

ARITHMETIC SEQUENCESIn an arithmetic sequence,

the difference between consecutive terms is constant.

This difference is called the common difference.

The common difference can be positive (the terms of the sequence are increasing in value)

or negative (the terms of the sequence are decreasing in value).

EXAMPLE:Is the given sequence arithmetic? If so, identify the

common difference.a.) 2, 4, 8, 16,…

b.) 6, 12, 18, 24,…

+2 +4+8

There is no common difference.

It is not an arithmetic sequence.

+6 +6 +6

There is a common difference, thereforethis is an arithmetic sequence.

The common difference is 6.

You can use an explicit formula to find the value of the nth term of an arithmetic sequence when the

previous term is unknown.

ARITHMETIC SEQUENCE FORMULAS

Recursive Formula:

Explicit Formula:

1 1a given value, n na a a d

1 1na a n d

In these formulas, an is the nth term, a1 is the 1st term, n is the number of the term, and d is the common difference.

EXAMPLE: • Suppose you participate in a bike-a-thon for charity.• The charity starts with $1100 in donations.• Each participant must raise at least $35 in pledges. • What is the minimum amount of money raised if there are 75

participants?

Find the 76th term of the sequence, 1100, 1135, 1170,…

Use the explicit formula 1 1na a n d Substitute a1 = 1100, n =7 6,

& d = 35 76 1100 76 1 35a

76 1100 75 35a

76 1100 2625a 76 3725a

With 75 participants, the bike-a-thon will raise a minimum of $3,725

Why the 76th term?

EXAMPLE:• Use the explicit formula to find the 25th term on

the sequence 5, 11, 17, 23, 29

Use the explicit formula 1 1na a n d Substitute a1 = 5, n =25,

& d = 6 25 5 25 1 6a

25 5 24 6a

25 5 144a 25 149a

ARITHMETIC MEANThe arithmetic mean of any two numbers is the

average of the two numbers

For any three sequential terms in an arithmetic sequence, the middle term is the arithmetic mean

of the first and third terms.

You can use arithmetic mean to find

a missing term of an arithmetic sequence.

sum of two numbersArithmetic Mean =

2

EXAMPLEFind the missing term of each arithmetic

sequence.a.)84, ? , 110 b.)

84 110=

2

194=

2

= 97

24, ? , 57

24 57=

2

81=

2

= 40.5


2


2

HOMEWORK

Worksheet: Arithmetic Sequences

GEOMETRIC SEQUENCES

GEOMETRIC SEQUENCES

The sequence 1, 2, 4, 8,… is a geometric sequence.

How would you describe the sequence?

Multiply by 2

The sequence 900, 300, 100,… is also a geometric sequence.

How would you describe the sequence?

Divide by 3 or Multiply by 1/3

GEOMETRIC SEQUENCES

What is common between geometric sequences?

In geometric sequences, the ratio between consecutive terms is constant.

This ratio is called the common ratio.

EXAMPLE:Is the given sequence geometric? If so, identify

the common ratio.a.) 5, 15, 45, 135,…

b.) 15, 30, 45, 60,…

x 3 x 3 x 3

Each term is being multiplied by 3.

So, yes, it is a geometric sequence.

The common ratio is3

x 2 x 1.5x 1.3

No common ratio; therefore, not a geometric sequence.

GEOMETRIC SEQUENCE FORMULAS

Recursive Formula:

Explicit Formula:

1 1a given value, n na a a r

11

nna a r

In these formulas, an is the nth term, a1 is the 1st term, n is the number of the term, and r is the common ratio.

EXAMPLE: • Suppose you want a reduced copy of a photograph.• The actual length of the photograph is 10 in.• The smallest size the copier can make is 64% of the original.• Find the length of the photograph after five reductions of 64%.

For five reductions, you need to find the 6th term of the geometric sequence 10, 6.4…

Use the explicit formula1

1n

na a r Substitute a1 = 10, n = 6,

& r = 0.64

6 16 10 0.64a

56 10 0.64a

6 10 0.1074a 6 1.07a

After five reductions of 64%, the photograph is about 1 in. long

GEOMETRIC MEAN

You can find the geometric mean of any two positive numbers by taking the positive square

root of the product of the two numbers.

You can use the geometric mean to find a missing term of a geometric sequence.

Geometric Mean = product of two #'s

EXAMPLEFind the missing term of each geometric

sequence.a.)12, ? , 3 b.)

G.M. = product of two #'s

= 12 3

= 36

= 6

5, ? , 2.8125

G.M. = product of two #'s

= 5 2.8125

= 14.0625

= 3.75

HOMEWORK

Worksheet: Geometric Sequences

ARITHMETIC SERIES – DAY 1

SERIES• A series is the expression for the sum of the

terms of a sequence• Finite sequences and series have terms that you

can count individually from 1 to a whole number n.• Infinite sequences and series continue without

end.Finite Sequence Finite Series

Infinite Sequence Infinite Series

6, 9, 12, 15, 18 6 + 9 + 12 + 15 + 18

3, 7, 11, 15, … 3 + 7 + 11 + 15 + …

EXAMPLE:Write the related series for each finite sequence. Then evaluate

the series.

a.) 2, 11, 20, 29, 38, 47Related Series:2 + 11 + 20 + 29 + 38 + 47

To evaluate, add the series

The sum of the terms of the sequence= 147

b.) 0.3, 0.6, 0.9, 1.2, 1.5, 1.8, 2.1, 2.4, 2.7, 3.0

Related Series:0.3 + 0.6 + 0.9 + 1.2 + 1.5 + 1.8 + 2.1 + 2.4 + 2.7 + 3.0

The sum of the terms of the sequence= 16.5

c.) 100, 125, 150, 175, 200, 225

Related Series:100 + 125 + 150 + 175 + 200 + 225

The sum of the terms of the sequence= 975

ARITHMETIC SERIES

• An arithmetic series is a series of terms whose terms form an arithmetic sequence.

• When a sequence has many terms, or when you know only the first & last terms of the sequence, you can use a formula to evaluate the related series quickly.

SUM OF A FINITE ARITHMETIC SERIES

The sum Sn of a Finite Arithmetic Series a1 + a2 + a3 + … + an is

12n n

nS a a

where a1 is the first term, an is the nth term, and n is the number of terms.

EXAMPLE:• Several rows of cross-stitches make up the

green roof in the picture to the right.

a.) Find the total number of green cross-stitches

in the roof.xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Sum of a finite arithmetic series 12n n

nS a a

Substitute n = 5, a1 = 5, & an = 13

There are 45 stitches in the green roof.

Row 1 = 5 stitches,Row 2 = 7 stitches, …

55 13

2nS 518

2 45

The # of cross-stitches in each row form a series.

HOMEWORK

Worksheet: Arithmetic Series – Day 1

ARITHMETIC SERIES – DAY 2

SUMMATION NOTATIONYou can use the summation symbol ∑ to write a

series.

Then you can use limits to indicate how many terms you are adding.

Limits are the least and greatest integral values of n.

3

1

5 1n

n

explicit formulafor the sequence

upper limit,greatest value of n

lower limit,least

value of n

EXAMPLE:Use summation notation to write each series for the

specified number of terms.

a.) 3 + 6 + 9 + … for 33 terms

Write the explicit formula 1 1na a n d Substitute a1 = 3 & d = 3

n = the # of the term, soleave that blank

3 1 3na n

distribute & simplify3 3 3na n 3na n

upper limit(greatest value of n)= 33

lower limit(least value of n) = 1

33

1

3n

n

CONTINUED…

b.) 1 + 2 + 3 + … ; n = 6



1 1 1na n

distribute & simplify1 1na n

na nupper limit

(greatest value of n)= 6


6

1n

n

CONTINUED…

c.) 3 + 8 + 13 + 18 … ; n = 9



3 1 5na n

distribute & simplify3 5 5na n 5 2na n

upper limit(greatest value of n)

= 9


9

1

5 2n

n

EXAMPLE:For each sum, find the number of terms, the first term, and the

last term. Then evaluate the series.

a.) 3

1

5 1n

n

Step 1: Find the number of terms

Since the values of n are 1, 2, & 3, there are 3 terms in the series

Step 2: Find the 1st & last terms

1st term: 5 1n 5 1 1 6last term:5 1n 5 3 1 16

Step 3: Evaluate Use formula 12n n

nS a a

12n n

nS a a 3

6 162

322

2 33

CONTINUED…

b.) 10

1

3n

n

Step 1: Find the number of terms

Since the values of n are 1 thru 10, there are 10 terms in the series


1st term: 3n 1 3 2last term: 3n 10 3 7


nS a a

12n n

nS a a 10

2 72

5 5 25

CONTINUED…

c.) 5

2

2n

n Step 1: Find the number of terms

Since the values of n are 2 thru 5, there are 4 terms in the series


1st term: 2n 22 4

last term: 2n 25 25


nS a a

12n n

nS a a 4

4 252

2 29 58

HOMEWORK

Worksheet: Arithmetic Series - Day 2

GEOMETRIC SERIES – DAY 1Evaluating a Finite Geometric Series

GEOMETRIC SERIES

A geometric series is the expressions for the sum

of the terms of a geometric sequence.

SUM OF A FINITE GEOMETRIC SERIES

The sum Sn of a Finite Geometric Series a1 + a2 + a3 + … + an, r ≠ 1, is

1 1

1

n

n

a rS

r

where a1 is the first term, r is the common ratio, and n is the number of terms.

EXAMPLE:Evaluate the series to the given term.a.) 3 + 6 + 12 + 24…; S6

Use the formula 1 1

1

n

n

a rS

r

Substitute a1 = 3, r = 2, & n = 6

6

6

3 1 2

1 2S

Simplify 3 1 64

1

3 63

1

189

1

189

CONTINUED…

b.) -45 + 135 - 405 +…; S5

Use the formula 1 1

1

n

n

a rS

r

Substitute a1 = -45, r = -3, & n = 5

5

5

45 1 ( 3)

1 ( 3)S

Simplify 45 1 ( 243)

4

45 244

4

10980

4

2745

EXAMPLE: • In March, the Floyd family starts saving for a vacation in

August, which they expect to cost $1375.• They start with $125 and each month they plan to

deposit 20% more than the previous month. • Will they have enough money for their trip?

Use the formula 1 1

1

n

n

a rS

r

Substitute a1 = 125,

r = 1.2, & n = 6

6125 1 1.2

1 1.2

Simplify

125 1 2.986

0.2

125 1.986

0.2

248.25

0.2

$1,241.25

No, the Floyd family will not have enough money for their vacation.

HOMEWORK

Worksheet: Geometric Series - Day 1

GEOMETRIC SERIES – DAY 2Evaluating an Infinite Geometric Series

INFINITE GEOMETRIC SERIES

In some cases, you can evaluate an infinite geometric series.

When , the series converges, or gets closer and closer, to the sum S. It will eventually have a

sum.

When , the series diverges, or approaches no limit.

1r

1r

EXAMPLE:Decide whether each infinite geometric series diverges or

converges. State whether the series has a sum.

a.)

b.)

1 11 ...

3 9 a1 = 1 & a2 = -1/3 therefore, r =

1

3

Since , the series converges and the series has a sum 1r

1

1

5 2n

n

a1 = 5(2)1-1= 5(2)0= 5(1)= 5

a2 = 5(2)2-1= 5(2)1= 5(2)= 10Since a1 = 5 and a2 = 10,

r = 2

Since , the series diverges and the series does not have a sum 1r

CONTINUED…Decide whether each infinite geometric series diverges or

converges. State whether the series has a sum.

c.)

d.)

1 11 ...

5 25 a1 = 1 & a2 = 1/5 therefore, r =

1

5Since , the series converges and the series has a sum 1r

4 8 16 .... a1 = 4 & a2 = 8

Since , the series diverges and the series does not have a sum 1r

therefore, r = 2

SUM OF AN INFINITE GEOMETRIC SERIES

An infinite geometric series with converges to the sum

1

1

aS

r

1r

Where a1 is the first term and r is the common ratio.

EXAMPLE:Evaluate each infinite geometric series

a.) 1 1 1

1 ...2 4 8

1

1

aS

r

1

11

2

112

2

b.)3 3 3

3 ...2 4 8

1

1

aS

r

3

11

2

31

12

2

EXAMPLE:• The length of the outside shell of each closed chamber

of a chambered nautilus is 0.9 times the length of the larger chamber next to it.

• Estimate the total length of the outside shell for the enclosed chambers.

The outside edge of the largest enclosed chamber is 27 mm long, so a1 = 27

Use the formula 1

1

aS

r

Substitute

a1 = 27 & r = 0.927

1 0.9

Simplify 27

0.1

270

HOMEWORK

Worksheet: Geometric Series – Day 2

U NIT 7 Sequences & Series. P ATTERNS E XAMPLE : o Suppose every student in class has a phone...

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