Arithmetic Sequence - MOORE'S MATH · 2019. 8. 9. · AFM Notes 4-1 Arithmetic Sequence 2 Example...

AFM Notes Name: _________________________________ 4 – 1 Arithmetic Sequence Date: _________________ Block: _________

1

Arithmetic Sequence

An ________________________________ ____________________________ is a

sequence of numbers in which each __________________________ after the first

term is found by adding the _____________________________

___________________________ to the preceding term.

nth Term of an Arithmetic Sequence

€

an = a1 + n −1( )d

€

an = _________________________________

€

a1 = _________________________________

€

n = _________________________________

€

d = _________________________________

Example 1:

Find the next four terms of the arithmetic sequence 7, 11 , 15 ,…..

Step 1: Find the common difference by subtracting two consecutive terms.

Step 2: Now add ____________, the common difference, to the third term to get the

4th term. Repeat this step 3 more times to find the next 4 terms.

AFM Notes 4-1 Arithmetic Sequence

2

Example 2:

Find the thirteenth term of the arithmetic sequence with

€

a1 = 21 and

€

d = −6 .

Use the formula for the nth term of an arithmetic sequence using

€

a1 = 21,

€

n =13,

and

€

d = −6.

Example 3:

Write an equation for the nth term of the arithmetic sequence In this sequence

€

a1 = −14 and

€

d = 9. Use the formula for

€

an to write an equation.

YOU TRY: Find the next four terms of each arithmetic sequence.

1. 106, 111, 116, …

2. -28, -31, -34, …

3. 207, 194, 181, …


3

Find the first 5 terms of each arithmetic sequence described.

4.

€

a1 =101,

€

d = 9

5.

€

a1 = −60,

€

d = 4

6.

€

a1 = 210,

€

d = −40

Find the indicated term of each arithmetic sequence.

7.

€

a1 = 4 ,

€

d = 6,

€

n =14

8.

€

a1 = −4 ,

€

d = −2 ,

€

n =12

9.

€

a1 = 80,

€

d = −8 ,

€

n = 21

10.

€

a10 for 0, -3, -6, -9, …

Write an equation for the nth term of each arithmetic sequence.

11. 18, 25, 32, 39, …

12. -110, -85, -60, -35, …

13. 6.2, 8.1, 10.0,11.9, …


4

Arithmetic Mean

The ________________________________ __________________________ of an

arithmetic sequence are the terms between any two nonsuccessive terms of the

sequence. To find the ___________ arithmetic means between two terms of a

sequence, use the following steps.

Step 1: Let the two terms given be _______ and ________, where ________________.

Step 2: Substitute in the formula _____________________________________________.

Step 3: Solve for _____________, and use that value to find the ________ arithmetic

means: ______________________________________________________________

Example 4:

Find the five arithmetic means between 37 and 121.

You can use the nth term formula to find the _________________________

________________________. In the sequence 37 , ? , ? , ? , ? , ? , 121 , … ,

€

a1is

37 and

€

a7is 121.


5

YOU TRY: Find the arithmetic means in each sequence.

1. 5 , ? , ? , ? , -3

2. 16 , ? , ? , 37

3. 108 , ? , ? , ? , ? , 48

4. -14 , ? , ? , ? , -30

5. 61 , ? , ? , ? , ? , 116

6. -40 , ? , ? , ? , ? , ? , -82

AFM Notes Name: _________________________________ 4 – 2 Arithmetic Series Date: _________________ Block: _________

6

Arithmetic Series

The sum ____________ of the first n terms of an _________________________________

_______________________ is give by the formula:

€

Sn =n22a1 + n −1)d( )[ ] or

€

Sn =n2a1 + an( )

€

Sn = _________________________________

€

n = _________________________________

€

a1 = _________________________________

€

d = _________________________________

€

an = _________________________________

Example 1:

Find Sn for the arithmetic series with a1 = 14, an = 101, and n = 30.

Use the sum formula for an arithmetic series.

Example 2:

Find the sum of all positive odd integers less than 180.

The series 1 + 3 + 5 + … + 179.

Find n using the formula for the nth term of an arithmetic sequence.

AFM Notes 4-2 Arithmetic Series

7

YOU TRY: Find the Sn for each arithmetic series described.

1.

€

a1 =12,an =100,n =12

2.

€

a1 = 50,an = −50,n =15

3.

€

a1 = 20,d = 4,an =112

4.

€

a1 = −8,d = −7,an = −71

5.

€

a1 = 42,n = 8,d = 6

6.

€

a1 = 4,n = 20,d = 2 12


8

Find the sum of each arithmetic series. 7. 8 + 6 + 4 + … + (−10)

8. 16 + 22 + 28 + … + 112

9. −45 + (−41) + (−37) + … + 35 Find the first three terms of each arithmetic series described.

10.

€

a1 =12,an =174,Sn =1767

11.

€

a1 = 80,an = −115,Sn = −245

12.

€

a1 = 6.2,an =12.6,Sn = 84.6


9

Sigma Notation

A shorthand notation for representing a series makes use of the Greek letter

______________. The _________________________ ____________________________ for

the series 6 + 12 + 18 + 24 + 30 is

€

6nn=1

5

∑ .

Example 3:

Evaluate

€

3k + 4( )k=1

18

∑

YOU TRY: Find the sum of each arithmetic series.

1.

€

2n +1( )n=1

20

∑

2.

€

x −1( )x=5

25

∑

3.

€

2r − 200( )r=10

75

∑

4.

€

n −100( )n=20

85

∑

5.

€

25 − 2 j( )j=12

32

∑

6.

€

3n + 4( )n=20

50

∑

AFM Notes Name: _____________________________ 4 – 3 Geometric Sequence Date: ________________ Block: _______

10

Geometric Sequence

A ____________________________ __________________________ is a sequence in

which each term after the first is the _________________________ of the previous

term and a constant called the ________________________ ______________________.

nth Term of a Geometric Sequence

€

an = a1⋅ rn−1

€

an = _________________________________

€

a1 = _________________________________

€

r = _________________________________

€

n = _________________________________

Example 1:

Find the next two terms of a geometric sequence 1200 , 480 , 192 , … ,

Example 2:

Write an equation for the nth term of the geometric sequence 3.6, 10.8, 32.4, …

AFMNotes 4-3GeometricSequence

11

YOU TRY: Find the next two terms of each geometric sequence.

1. 6 , 12 , 24 , …

2. 180 , 60 , 20 , …

3. 2000 , −1000 , 500 , …

4. 0.8 , −2.4 , 7.2 , …

5. 80 , 60 , 45 , …

6. 3 , 16.5 , 90.75 , …

Find the first fiver terms of each geometric sequence described.

7.

€

a1 =19,r = 3

8.

€

a1 = 240,r = −34

9.

€

a1 =10,r =52


12

Find the indicated term of each geometric sequence. 10.

€

a1 = −10,r = 4,n = 2

11.

€

a1 = −6,r = −12,n = 8

12.

€

a3 = 9,r = −3,n = 7

13.

€

a4 = −54,r = −3,n = 6

Write an equation for the nth term of each geometric sequence. 14. 500 , 350 , 245 , …

15. 8 , 32 , 128 , …

16. 11 , −24.2 , 53.24 , …


13

The ____________________________ __________________ of a geometric sequence

are the terms between any two _____________________________________ terms of a

the sequence.

To find the _________ geometric means between two terms of a sequence use the

following steps.

Step 1: Let the two terms give be ______ and ______, where ____________________.

Step 2: Substitute in the formula _____________________________________________.

Step 3: Solve for __________, and use that value to find the k geometric means:

_____________________________________________________________________

Example 3:

Find the three geometric means between 8 and 40. 5.

Use the nth term formula to find the value of r. In the sequence

8 , ? , ? , ?, 40.5 , a1 is 8 and a5 is 40.5.

YOU TRY: Find the geometric means in each sequence.

1. 5 , ? , ? , ?, 405

2. 5 , ? , ? , 20.48

3.

€

35

, ? , ? , ?, 375


14

4. -24, ? , ? ,

€

19

5. 12 , ? , ? , ? ? , ?,

€

316

6. 200 , ? , ? , ?, 414.72

7.

€

3549

, ? , ? , ?, ?, -12,005

8. 4 , ? , ? , ?, 156

€

14

9. −

€

181

, ? , ? , ?, ?, ?, -9

10. 100 , ? , ? , ?, 384.16

AFM Notes Name: _________________________________ 4 – 4 Geometric Series Date: _________________ Block: _________

15

A _______________________________ _________________________ is the indicated sum

of consecutive terms of a geometric sequence.

Sum of a Geometric Sequence

€

Sn =a1(1− r

n )1− r

or

€

Sn =a1 − anr1− r where

€

r ≠1.

€

Sn = _________________________________

€

n = _________________________________

€

a1 = _________________________________

€

r = _________________________________

€

an = _________________________________

Example 1:

Find the sum of the first four terms of the geometric sequence for which

€

a1 =120 and

€

r =13

Example 2:

Find the sum of the geometric series

€

4⋅ 3 j−2j=1

7

∑

AFM Notes 4-4 Geometric Series

16

YOU TRY: Find Sn for each geometric series described.

1.

€

a1 = 2,an = 486,r = 3

2.

€

a1 =1200,an = 75,r =12

3.

€

a1 =125,an =125,r = 5

4.

€

a1 = 2,r = 6,n = 4

5.

€

a3 = 20,a6 =160,n = 8

6.

€

a4 =16,a7 =1024,n =10


17

Find the sum of each geometric series. 7. 6 + 18 + 54 +…𝑡𝑜 6 terms

8.

€

14

+12

+1… go 10 terms

9.

€

2 j

j=4

8

∑

10.

€

3⋅ 2k−1k=1

7

∑


18

You can use one of the formulas for the sum of a geometric series to help find a particular term of the series. Example 3:

Find a1 in the geometric series for which S6 = 441 and r = 2.

Example 4:

Find a1 in the geometric series for which Sn = 244, an = 324, and r= - 3. Since you do not know the value of n, use the alternate sum formula. Example 5:

Find a4 in the geometric series for which Sn = 796.875,

€

r =12

, and n= 8.

First use the sum formula to find

€

a1.


19

YOU TRY:

Find the indicated term for each geometric series described.

1.

€

Sn = 726,an = 486,r = 3;a1

2.

€

Sn = 850,an =1280,r = −2;a1

3.

€

Sn =1023.75,an = 512,r = 2;a1

4.

€

Sn =118.125,an = −5.625,r = −12;a1

5.

€

Sn =183,r = −3,n = 5;a1

6.

€

Sn =1705,r = 4,n = 5;a1

AFM Notes Name: _____________________________ 4 – 5 Infinite Geometric Series Date: _________________ Block: ______

20

A geometric series that does not end is called an ________________________

_________________________ ____________________. Some infinite geometric series

have sums, but others do not because the ___________________________

________________ increase without approaching a limiting value.

Sum of an Infinite Geometric Series

€

S =a11− r for

€

−1 < r <1.

If

€

r ≥1, the infinite geometric series does not have a sum.

€

S = _________________________________

€

a1 = _________________________________

€

r = _________________________________

Example 1:

Find the sum of each infinite geometric series, if it exists.

a) 75 + 15 + 3 + …

First, find the value of r to determine if the sum exists.

b)

€

48 − 13

⎛

⎝ ⎜

⎞

⎠ ⎟ n−1

n=1

∞

∑

AFM Notes 4-5 Infinite Geometric Series

21

YOU TRY:


1.

€

a1 = −7,r =58

2.

€

1+54

+2516

+ ...

3.

€

a1 = 4,r =12

4.

€

18 − 9 + 4 12− 2 14

+ ...

5.

€

50 45⎛

⎝ ⎜ ⎞

⎠ ⎟ n−1

n=1

∞

∑

6.

€

22 − 12

⎛

⎝ ⎜

⎞

⎠ ⎟ k−1

k=1

∞

∑


22

A repeating decimal represents a fraction. To find the fraction, write the decimal

as an infinite geometric series and use the formula for the sum.

Example 2: Write each repeating decimal as a fraction.

Write the repeating decimal as a sum for:

a)

€

0.42

b)

€

0.524

YOU TRY:

Write each repeating decimal as a fraction.

7.

€

0.2

8.

€

0.62


23

9.

€

0.06

10.

€

0.0138

11.

€

0.245

Example 3:

A patient is given a 20mg injection of a therapeutic drug. Each day, the patient’s

body metabolizes 50% of the drug present, so that after 1 day only one-half of the

original amount remains, after 2 days only one-fourth remains, and so on. The

patient is given a 20 mg injection of the drug every day at the same time.

a) Write a geometric series that gives the drug level in this patient’s body

right after the nth injection

Q1 =

Q2 =

Q3 =

Q4 =

Qn =


24

Note that this is an example of a geometric series, with a1 = _________________ and

r = _____________.

b) Write this series using sigma notation.

c) Write the explicit formula for this geometric series.

d) Suppose this patient receives injections over a long period of time. What

will happen to the drug level in the patient’s body?

Q10 =

Q15 =

Q25 =


25

The Sum of an Infinite Geometric Series

As the number of terms in a geometric series approaches ____________, if the sum

of the terms approaches a finite number we say the series ______________. Note

that this can only happen if _______________. (Or more specifically, if

_______________.)

We can write a formula for this sum.

If |r| < 1, the sum of an infinite geometric series is:

If |r| ≥ 1, the series does not converge; instead we say it _________________.

Example 4:


a)

€

2 34⎛

⎝ ⎜ ⎞

⎠ ⎟ n−1

n=1

∞

∑

b)

€

2 2( )n−1n=1

∞

∑

c) 2 + 5 + 8 + 11 +…

d) 16 + 8 + 4 + 2 +…

AFM Notes Name: _____________________________ 4 – 6 Recursion and Special Sequences Date: _________________ Block: ______

26

In a ___________________________ ________________________, each succeeding term

is formulated from one or more _____________________ terms. A ________________

____________________________ for a sequence has two parts.

1. the ____________________(s) of the _________________ ____________(s), and

2. an ______________________ that shows how to find each _________________

from the _______________________(s) before it.

Example 1:

Find the first five terms of the sequence in which

€

a1 = 6,

€

a2 =10and

€

an = 2an−2 for

€

n ≥ 3.

You Try:

Find the first five terms of each sequence.

1.

€

a1 =1,a2 =1,an = an−1 + an−2( ),n ≥ 3

2.

€

a1 =1,an =1

1+ an−1

⎛

⎝ ⎜

⎞

⎠ ⎟ ,n ≥ 2

3.

€

a1 = 3,an = an−1 + 2 n − 2( ),n ≥ 2

AFM Notes 4-6 Recursion and Special Sequences

27

4.

€

a1 = 5,an + an−1 + 2,n ≥ 2

5.

€

a1 =1,,an = n −1( )an−1,n ≥ 2

6.

€

a1 = 7,an = 4an−1 −1,n ≥ 2

7.

€

a1 = 3,a2 = 4,an = 2an−2 + 3an−1,n ≥ 3

8.

€

a1 = 0.5,an = an−1 + 2n,n ≥ 2

9.

€

a1 = 8,a2 =10,an =an−2an−1

,n ≥ 3

10.

€

a1 =100,an =an−1n,n ≥ 2


28

Combining composition of functions with the concept of recursion

leads to the process of _______________________. Iteration is the

process of ________________________ a function with _________________

repeatedly.

Example 2:

Find the first three iterates of

€

f (x) = 4x − 5for an initial value of

€

x0 = 2.

You Try:

Find the first three iterate of each function for the given initial value.

1.

€

f (x) = x −1;x0 = 4

2.

€

f (x) = x 2 − 3x;x0 =1

3.

€

f (x) = x 2 + 2x +1;x0 = −2


29

4.

€

f (x) = 4x − 6;x0 = −5

5.

€

f (x) = x 3 − 5x 2;x0 =1

6.

€

f (x) = 4x 2 − 9;x0 = −1

7.

€

f (x) =x −1x + 2

;x0 =1

8.

€

f (x) =12x +11( );x0 = 3

9.

€

f (x) =3x;x0 = 9

10.

€

f (x) = x − 1x;x0 = 2

11.

€

f (x) = x 3 − 5x 2 + 8x −10;x0 =1

Arithmetic Sequence - MOORE'S MATH · 2019. 8. 9. · AFM Notes 4-1 Arithmetic Sequence 2 Example...

Documents

Transcript of Arithmetic Sequence - MOORE'S MATH · 2019. 8. 9. · AFM Notes 4-1 Arithmetic Sequence 2 Example...