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Arithmetic Sequences, Geometric Sequences, & Scatterplots

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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.
A. B.
C. D.
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
A.
B.
C.
D.
Concept
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
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
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
Example 3
an = 1 + (n – 1)(9) a1 = 1, d = 9
an = 1 + 9n – 9 Distributive Property
an = 9n – 8 Simplify.
Find the nth Term
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
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
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 +11
= 12 + (n – 1)11 a1 = 12 and d = 11
= 12 + 11n – 11 Distributive Property
= 11n + 1 Simplify.
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
= –8 __ 1 –8 ___ 64
–8 = –8 = –8 ______ –512
Example 2
Find Terms of Geometric Sequences
Step 2 Multiply each term by the common ratio to find
the next three terms.
× (–8) × (–8) × (–8)
Answer: The next 3 terms in the sequence are 4096; –32,768; and 262,144.
–32,768 4096 –512
Find Terms of Geometric Sequences
B. Find the next three terms in the geometric sequence. 40, 20, 10, 5, ....
40 20 10 5
= __ 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 4
__ 5 8
× __ 1 2
× __ 1 2
× __ 1 2
Answer: The next 3 terms in the sequence are , __ 5 2
__ 5 4
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
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
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
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.
y – y1 = m(x – x1)
y – 1000 ≈ 35.1x – 64,870
y – 1000 = (x – 1850)
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

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