Arithmetic Sequences, Geometric Sequences, & Scatterplots

Review for EOC Arithmetic Sequences, Geometric Sequences, & Scatterplots

Over Lesson 3–4

5-Minute Check 1

What is the constant of variation for the equation of the line that passes through (2, –3) and (8, –12)?

A.

B.

C.

D.

Over Lesson 3–4

5-Minute Check 2

Which graph represents y = –2x?

A. B.

C. D.

Over Lesson 3–4

5-Minute Check 3

A. 6

B. 8

C. 24

D. 16

Suppose y varies directly with x. If y = 32 when x = 8, find x when y = 64.

Over Lesson 3–4

5-Minute Check 5

Which direct variation equation includes the point (–9, 15)?

A.

B.

C.

D.

Concept

Example 1

Identify Arithmetic Sequences

A. Determine whether –15, –13, –11, –9, ... is an arithmetic sequence. Explain.

Answer: This is an arithmetic sequence because the difference between terms is constant.

Example 1

Identify Arithmetic Sequences

Answer: This is not an arithmetic sequence because the difference between terms is not constant.

B. Determine whether is an arithmetic sequence. Explain.

Example 1 CYP A

A. Determine whether 2, 4, 8, 10, 12, … is an arithmetic sequence.

A.  cannot be determined

B.  This is not an arithmetic sequence because the difference between terms is not constant.

C.  This is an arithmetic sequence because the difference between terms is constant.

Example 1 CYP B

B. Determine whether … is an

arithmetic sequence.

A.  cannot be determined

B.  This is not an arithmetic sequence because the difference between terms is not constant.

C.  This is an arithmetic sequence because the difference between terms is constant.

Example 2

Find the Next Term

Find the next three terms of the arithmetic sequence –8, –11, –14, –17, ….

Find the common difference by subtracting successive terms.

The common difference is –3.

Example 2

Find the Next Term

Subtract 3 from the last term of the sequence to get the next term in the sequence. Continue subtracting 3 until the next three terms are found.

Answer: The next three terms are –20, –23, and –26.

Example 2 CYP

A. 78, 83, 88

B. 76, 79, 82

C. 73, 78, 83

D. 83, 88, 93

Find the next three terms of the arithmetic sequence 58, 63, 68, 73, ….

Concept

Example 3

The common difference is 9.

A. Write an equation for the nth term of the arithmetic sequence 1, 10, 19, 28, … .

In this sequence, the first term, a1, is 1. Find the common difference.

Step 1 Find the common difference.

Find the nth Term

Step 2 Write an equation.

Example 3

an = a1 + (n – 1)d Formula for the nth term

an = 1 + (n – 1)(9) a1 = 1, d = 9

an = 1 + 9n – 9 Distributive Property

an = 9n – 8 Simplify.

Find the nth Term

Example 3

Check For n = 1, 9(1) – 8 = 1.

For n = 2, 9(2) – 8 = 10.

For n = 3, 9(3) – 8 = 19, and so on.

Answer: an = 9n – 8

Find the nth Term

Example 3

B. Find the 12th term in the sequence.

Replace n with 12 in the equation written in part A.

an = 9n – 8 Formula for the nth term

a12 = 9(12) – 8 Replace n with 12.

a12 = 100 Simplify.

Answer: a12 = 100

Find the nth Term

Example 4 A

Arithmetic Sequences as Functions

NEWSPAPERS The arithmetic sequence 12, 23, 34, 45, ... represents the total number of ounces that a bag weighs after each additional newspaper is added. A. Write a function to represent this sequence.

12 23 34 45

+11

The common difference is 11.

+11 +11

Example 4 A

Arithmetic Sequences as Functions

an = a1 + (n – 1)d Formula for the nth term

= 12 + (n – 1)11 a1 = 12 and d = 11

= 12 + 11n – 11 Distributive Property

= 11n + 1 Simplify.

Answer: The function is an = 11n + 1.

Example 1

Identify Geometric Sequences

A. Determine whether the sequence is arithmetic, geometric, or neither. Explain. 0, 8, 16, 24, 32, ...

0 8 16 24 32

8 – 0 = 8

Answer: The common difference is 8. So, the sequence is arithmetic.

16 – 8 = 8 24 – 16 = 8 32 – 24 = 8

Example 1

Identify Geometric Sequences

B. Determine whether the sequence is arithmetic, geometric, or neither. Explain. 64, 48, 36, 27, ...

64 48 36 27

Answer: The common ratio is , so the sequence is geometric.

__ 3 4

__ 3 4

___ 48 64 =

__ 3 4

___ 36 48 =

__ 3 4

___ 27 36 =

Example 1

A. arithmetic

B. geometric

C. neither

A. Determine whether the sequence is arithmetic, geometric, or neither. 1, 7, 49, 343, ...

Example 1

B. Determine whether the sequence is arithmetic, geometric, or neither. 1, 2, 4, 14, 54, ...

A. arithmetic

B. geometric

C. neither

Example 2

Find Terms of Geometric Sequences

A. Find the next three terms in the geometric sequence. 1, –8, 64, –512, ...

1 –8 64 –512

The common ratio is –8.

= –8 __ 1 –8 ___ 64

–8 = –8 = –8 ______ –512

64

Step 1 Find the common ratio.

Example 2

Find Terms of Geometric Sequences

Step 2 Multiply each term by the common ratio to find

the next three terms.

262,144

× (–8) × (–8) × (–8)

Answer: The next 3 terms in the sequence are 4096; –32,768; and 262,144.

–32,768 4096 –512

Example 2

Find Terms of Geometric Sequences

B. Find the next three terms in the geometric sequence. 40, 20, 10, 5, ....

40 20 10 5

Step 1 Find the common ratio.

= __ 1 2

___ 40 20 = __ 1

2 ___ 10 20 = __ 1

2 ___ 5 10

The common ratio is . __ 1 2

Example 2

Find Terms of Geometric Sequences

Step 2 Multiply each term by the common ratio to find

the next three terms.

5 __ 5 2

__ 5 4

__ 5 8

× __ 1 2

× __ 1 2

× __ 1 2

Answer: The next 3 terms in the sequence are , __ 5 2

__ 5 4

, and . __ 5 8

Example 2

A. Find the next three terms in the geometric sequence. 1, –5, 25, –125, ....

A. 250, –500, 1000

B. 150, –175, 200

C. –250, 500, –1000

D. 625, –3125, 15,625

Example 2

B. Find the next three terms in the geometric sequence. 800, 200, 50, , .... __

2 25

A. 15, 10, 5

B. , ,

C. 12, 3,

D. 0, –25, –50

__ 3 4

__ 8 25 ____ 25

128 ___ 25 32

Concept

Example 3

Find the nth Term of a Geometric Sequence

A. Write an equation for the nth term of the geometric sequence 1, –2, 4, –8, ... . The first term of the sequence is 1. So, a1 = 1. Now find the common ratio. 1 –2 4 –8

= –2 ___ –2 1

= –2 ___ 4 –2

= –2 ___ –8 4

an = a1rn – 1 Formula for the nth term an = 1(–2)n – 1 a1 = 1 and r = –2

The common ratio is –2.

Answer: an = 1(–2)n – 1

Example 3

Find the nth Term of a Geometric Sequence

B. Find the 12th term of the sequence. 1, –2, 4, –8, ... . an = a1rn – 1 Formula for the nth term a12 = 1(–2)12 – 1 For the nth term, n = 12.

= 1(–2)11 Simplify. = 1(–2048) (–2)11 = –2048 = –2048 Multiply.

Answer: The 12th term of the sequence is –2048.

A.

B.

C.

D.

Example 3

A. Write an equation for the nth term of the geometric sequence 3, –12, 48, –192, ....

Example 3

A. 768

B. –3072

C. 12,288

D. –49,152

B. Find the 7th term of this sequence using the equation an = 3(–4)n – 1.

Concept

Example 1

Evaluate a Correlation

TECHNOLOGY The graph shows the average number of students per computer in Maria’s school. Determine whether the graph shows a positive correlation, a negative correlation, or no correlation. If there is a positive or negative correlation, describe its meaning in the situation.

Sample Answer: The graph shows a negative correlation. Each year, more computers are in Maria’s school, making the students-per-computer rate smaller.

Example 1

A. Positive correlation; with each year, the number of mail-order prescriptions has increased.

B. Negative correlation; with each year, the number of mail-order prescriptions has decreased.

C. no correlation

D. cannot be determined

The graph shows the number of mail-order prescriptions. Determine whether the graph shows a positive correlation, a negative correlation, or no correlation. If there is a positive or negative correlation, describe it.

Concept

Example 2

Write a Line of Fit

POPULATION The table shows the world population growing at a rapid rate. Identify the independent and dependent variables. Make a scatter plot and determine what relationship, if any, exists in the data.

Example 2

Write a Line of Fit

Step 1 Make a scatter plot. The independent variable is the year, and the dependent variable is the population (in millions). As the years increase, the population increases. There is a positive correlation between the two variables.

Example 2

Write a Line of Fit

Step 2 Draw a line of fit. No one line will pass through all of the data points. Draw a line that passes close to the points. A line of fit is shown.

Example 2

Write a Line of Fit

Step 3 Write the slope-intercept form of an equation for the line of fit. The line of fit shown passes through the points (1850, 1000) and (2004, 6400). Find the slope.

Slope formula

Let (x1, y1) = (1850, 1000) and (x2, y2) = (2004, 6400).

Simplify.

Example 2

Write a Line of Fit

y – y1 = m(x – x1)

y – 1000 ≈ 35.1x – 64,870

y – 1000 = (x – 1850)

y ≈ 35.1x – 63,870

Answer: The equation of the line is y = 35.1x – 63,870.

Use m = and either the point-slope form or the slope-intercept form to write the equation of the line of fit.

Example 2a

A. There is a positive correlation between the two variables.

B. There is a negative correlation between the two variables.

C. There is no correlation between the two variables.

D. cannot be determined

The table shows the number of bachelor’s degrees received since 1988. Draw a scatter plot and determine what relationship exists, if any, in the data.