Sequences A2/trig. Sequences: Vocabulary Sequence: an ordered list of numbers –Ex. -2, -1, 0, 1,...

Sequences

A2/trig

Sequences:Vocabulary

• Sequence: an ordered list of numbers– Ex. -2, -1, 0, 1, 2, 3

• Term: each number in a sequence– Ex. a1, a2, a3, a4, a5, a6

• Recursive Formula: finding the next term depends on knowing a term or terms before it.

• Explicit formula: defines the nth term of a sequence.

Vocabulary

• Recursive Formula: – Uses one or more previous terms to generate

the next term.

– Any term: an

– Previous term: an-1 an-1

Examples:

A) Write the first six terms of the sequence

where a1 = -2 and an = 2an-1 – 1

( Always list the terms with subscripts first:) a1, a2, a3, a4, a5, a6

B) Write the first six terms of the sequence where a1 = 4 and an = 3an-1 + 5

a1, a2, a3, a4, a5, a6

Examples:

A) Write the first six terms of the sequence

where a1 = -2 and an = 2an-1 – 1 a1, a2, a3, a4, a5, a6

-2,-5,-11,-23,-47,-95

B) Write the first six terms of the sequence where a1 = 4 and an = 3an-1 + 5

a1, a2, a3, a4, a5, a6

4, 17, 56 ,173, 524, 1577

Recursive formula depending on two previous terms:

• a1=-2, a2 = 3 ak= ak-1 + ak-2

• Find the first 6 terms.

Recursive formula depending on two previous terms:

• a1=-2, a2 = 3 ak= ak-1 + ak-2

• Find the first 6 terms.

• a1, a2, a3, a4, a5, a6

• -2, 3, 1, 4, 5, 9

Two special sequence types:

Arithmetic sequence:a sequence in which each term is found by adding

a constant, called the common difference (d), to the previous term. Geometric sequence: a sequence in

which each term is founds by multiplying a constant, (r), called a common ratio to the previous term.

Some sequences are neither of these!

Do now:

A) Find the 10th term of a1 = 7 and an = an-1 + 6

Recursive formula

Example 1:

A) Find the 10th term of a1 = 7 and an = an-1 + 6

7,13,19,25,31,37,43,49,55,61

Formula for the nth term

an = a1 + (n – 1)d

What term you are looking for

First term in the sequence


Common difference

Example:Find the 10th term of a1 = 7 and

an = an-1 + 6 Write the explicit formula

(recall a10 = 61)

an= 7+ 6(n-1)an = 7 + 6n – 6an = 6n + 1a10 = 6(10) +1 = 61

an = a1 + d(n – 1)

Vocabulary

• Arithmetic Sequence: – A sequence generated by adding “d” a constant

number to pervious term to obtain the next term.– This number is called the common difference.

• Start by asking, What is d? a2 – a1

3, 7, 11, 15, … d = 4

8, 2, -4, -10, … d = -6

Find the explicit formula for these examples:

• For Arithmetic Sequences, use the formula: – an = a1 + d(n – 1)

3, 7, 11, 15, … d = 4

8, 2, -4, -10, … d = -6

solutions:

an = a1 + d(n – 1)3, 7, 11, 15, … d = 4

an = 3 + 4(n – 1)an = 3 + 4n – 4

an = 4n – 1

8, 2, -4, -10, … d = -6

an = 8 + -6(n – 1)an = 8 + -6n + 6

an = -6n + 14

Examples when a1 is not given

A) Find the 10th term of the arithmetic sequence where a3 = -5 and a6 = 16

B) Find the 15th term of the arithmetic sequence where a5 = 7 and a10 = 22

C) Find the 12th term of the arithmetic sequence where a3 = 8 and a7 = 20


A) Find the 10th term of the arithmetic sequence where a3 = -5 and a6 = 16

16- -5 =21 6-3 = 3

A)an = -19 + 7(n – 1)

B)an = -19 + 7n – 7

C) an = 7n – 26

A)A10 = 7(10)-26=44

d =213=7 a1 =−19


B) Find the 15th term of the arithmetic sequence where a5 = 7 and a10 = 22

22-7 = 15 10-5 = 5

a1=-5 a2=-2 a3=1 a4=4 a5=7

B) an = -5 + 3(n – 1)

C) an = -5 + 3n – 3

D) an = 3n – 8

B)A15 = 3(15)-8=37

d =155=3


C) Find the 12th term of the arithmetic sequence where a3 = 8 and a7 = 20

D = 3 a1 = 2C)an = 2 + 3(n – 1)

D)an = 2 + 3n – 3

E) an = 3n – 1

C)A12 = 3(12)-1=35

Vocabulary

• Arithmetic Means:– Terms in between 2 nonconsecutive terms– Ex. 5, 11, 17, 23, 29 11, 17, 23 are the

arithmetic means between 5 & 29

Example 3:

A) Find the 4 arithmetic means between 10 & -30

B) Find the 5 arithmetic means between 6 & 60

Example 3:

A) Find the 4 arithmetic means between 10 & -30

10, 2, -6, -14, -22 -30

d =−30−10

5=−8

Example 3:

B) Find the 5 arithmetic means between 6 & 60

6, 15, 24, 33, 42, 51, 60

60−66

=9

Geometric Sequences

multiplying

Do Now:

• Find the 5th term of a1 = 8 and an = 3an-1


Do Now:


• 8, 24, 72, 216, 648


• 5, 10, 20, 40, 80, 160, 320

Vocabulary

• Geometric Sequence:– A sequence generated by multiplying a constant

ratio to the previous term to obtain the next term.– This number is called the common ratio.

• What is r?

2, 4, 8, 16, … r = 2

27, 9, 3, 1, … r = 1/3

2

1

a

ra

Explicit Formula for the nth term

an = a1rn-1


First term in the sequence


Common Ratio

Explicit geometric formulaan = a1rn-1

• Find the Explicit formula and the 5th term of a1 = 8 and an = 3an-1


Explicit geometric formula

• Find the explicit formula and the 5th term of a1 = 8 and an = 3an-1

• an = a1rn-1

• an = 8(3)n-1 a5 = 8(3)4 = 648


• an = a1rn-1

• an = a1rn-1 a7 = 5(2)6 = 320

Warm up1. Find the 8th term of the sequence defined by a1= –4 and an= an-1+ 2

2. Find the 12th term of the arithmetic sequence in which a4= 2 and a7= 6

3. Find the four arithmetic means between 6 and 26.

4. Find the 5th term on the sequence defined by a1= 2 andan= 2an-1.

Summation

4

1

12n

n

Series• Series: the sum of a sequence

– Sequence: 1, 2, 3, 4– Series: 1 + 2 + 3 + 4

• Summation Notation:

4

1

12n

nEnd number

Start number

Formula to use

Summation Notation - __________________ EX. (for the above series)

4

1

12n

n

= 2(1)-1 + 2(2)-1 + 2(3) -1 + 2(4) -1

= 1 + 3 +5 + 7

=16

Summation Properties

• For sequences ak and bk and positive integer n:

1 1

1) n n

k kk k

ca c a

1 1 1

2) n n n

k k k kk k k

a b a b

Summation Formulas

• For all positive integers n:

Constant Linear

Quadratic

1

n

k

c nc

1

( 1)

2

n

k

n nk

2

1

( 1)(2 1)

6

n

k

n n nk

Example 1:

A) Evaluate

B) Evaluate

6

1

2k

k

6

1

4k

k

Example 1:

A) Evaluate

B) Evaluate

2kk=1

6

∑ =2+4 +6+8+10 +12 =42

4 kk=1

6

∑ =4(1+2 + 3+ 4 +5 +6)=84

Extra Example:

• Evaluate

(2m2 +3m+2)

m=0

2

∑

Extra Example:

• Evaluate

(2m2 +3m+2)

m=0

2

∑ =(2(0)2 +3(0)+2)+ (2(12 ) +3(1)+2)+ (2(2)2 +3(2)+2)

=2 + 7 +16

= 25

Arithmetic Series

Sum of an arithmetic sequence

1, 4, 7,10,13

9,1,−7,−15

Arithmetic Sequences

ADDTo get next term

2, 4, 8,16, 32

9,−3,1,−1 / 3

Geometric Sequences

MULTIPLYTo get next term

Arithmetic Series

Sum of Terms

Geometric Series

Sum of Terms

Do Now: add the terms of the 4 series above

1, 4, 7,10,13

9,1,−7,−15

Arithmetic Sequences

ADDTo get next term

2, 4, 8,16, 32

9,−3,1,−1 / 3

Geometric Sequences

MULTIPLYTo get next term

Arithmetic Series

Sum of Terms

35

−12

Geometric Series

Sum of Terms

62

20 / 3

Do Now: add the terms of the 4 series above

Vocabulary

• An Arithmetic Series is the sum of an arithmetic sequence.

Formula for arithmetic series

Sn=

21 naa

n

Example 1:

A. Find the series 1, 3, 5, 7, 9, 11

B. Find the series 8, 13, 18, 23, 28, 33, 38

Example 1:

A. Find the series 1, 3, 5, 7, 9, 11

B. Find the series 8, 13, 18, 23, 28, 33, 38

sn =62(1+11)=36

sn =72(8 + 38)=161

Example 2:

A) Given 3 + 12 + 21 + 30 + …, find S25

B) Given 16, 12, 8, 4, …, find S11

Find the 25th and the 11th terms by finding the explicit formula first.

Example 2:

A) Given 3 + 12 + 21 + 30 + …, find S25

B) Given 16, 12, 8, 4, …, find S11

an =3+9(n−1)=9n−6a25 =9(25)−6 =219

Now apply series formula..

Example 2:

A) Given 3 + 12 + 21 + 30 + …, find S25

B) Given 16, 12, 8, 4, …, find S11

s25 25232192775

s11 =112(16 +−24)=−44

Example 3:

A) Evaluate

B) Evaluate

12

1(6 2 )

kk

21

1(5 4 )

kk

Vocabulary

• An Geometric Series is the sum of an geometric sequence.

Formula for geometric series

Sn=

r1

r1a

n

1

Example 1:

• Given the series 3 + 4.5 + 6.75 + 10.125 + …find S5

Example 1:

• Given the series 3 + 4.5 + 6.75 + 10.125 + …find S5

s5 a11−rn

1−r

⎛

⎝⎜

⎞

⎠⎟3

1−1.55

1−1.5

⎛

⎝⎜

⎞

⎠⎟

39.5625

Vocabulary of Sequences (Universal)

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

nth term of geometric sequence→ an =a1rn−1

sumof nterms of geometric sequence→ a1(1−rn)1−r

7

1 1 1Find S of ...

2 4 8

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

S

n=

a1(1−rn)⎡⎣

⎤⎦

1−r

7

1 1 1Find S of ...

2 4 8

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

1/2

7

x

NA

11184r

1 1 22 4

n1

n

a r 1S

r 1

x =

121− 1

2

⎛

⎝⎜

⎞

⎠⎟

7⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

1−12

=

121− 1

2

⎛

⎝⎜

⎞

⎠⎟

7⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

−12

127/128

Do Now:

• Evaluate

• Evaluate

4 (−5)k−1

k=1

7

∑

2+ 3k−1( )

k=1

6

∑

Do Now:

• Evaluate

• Evaluate

4 (−5)k−1

k=1

7

∑ =52,084

2+ 3k−1( )

k=1

6

∑ =2+31−1 +32−1 +33−1 +34−1 +35−1 +36−1

=366

Do Now:

• Evaluate

• Evaluate

17

14( 5)k

k

16

132 k

k

n

a1r

sum =52,084

sum 728

Vocabulary • An Infinite Geometric Series is a geometric

series with infinite terms.

Formula for a convergent infinite geometric series

S =

If r <1 then the _______ can be found (converges)

If r 1 then the _______ can’t be found (diverges)

)1(1

r

a

SUM

SUM≥

Examples :

1. Find the sum of the infinite geometric series 3 + 1.2 + 0.48 + 0.192 + …

a. Find the partial sum (S4)

b. Determine the ratio

2. Find the sum of the infinite geometric series 8 + 9.6 + 11.52 + 13.824 + …

a. Find the partial sum (S4)

b. Determine the ratio

Example 1:

A) Find the sum of the infinite geometric series 3 + 1.2 + 0.48 + 0.192 + …

r = .4

B) Find the sum of the infinite geometric series 8 + 9.6 + 11.52 + 13.824 + …

r= 1.2 so it is divergent

s =3

1−.4=5

Example 2:

• Find the sum of the infinite geometric series below:

11

1

3kk

Example 2:

• Find the sum of the infinite geometric series below:

• r = .3=13 s =

19

1−13

=16

Example 3:

A. Write 0.2 as a fraction in simplest form.

B. Write 0.04 as a fraction in simplest form.

Example 3:

A. Write 0.2 as a fraction in simplest form.

B. Write 0.04 as a fraction in simplest form.

2

9

4

99

Sequences A2/trig. Sequences: Vocabulary Sequence: an ordered list of numbers –Ex. -2, -1, 0, 1,...

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