Write algebraic expressions to determine any term in an arithmetic sequence. common difference Main Idea/Vocabulary

Identify Arithmetic Sequences State whether the sequence 23, 15, 7, –1, –9, … is arithmetic. If it is, state the common difference. Write the next three terms of the sequence. 23, 15, 7, –1, –9 Notice that 15 – 23 = –8, 7 – 15 = –8, and so on. –8 The terms have a common difference of –8, so the sequence is arithmetic. Continue the pattern to find the next three terms. –9, –17, –25, –33 –8 Answer: The next three terms are –17, –25, and –33. Example 1

State whether the sequence 29, 27, 25, 23, 21, … is arithmetic State whether the sequence 29, 27, 25, 23, 21, … is arithmetic. If it is, state the common difference. Write the next three terms of the sequence. A. arithmetic; –2; 19, 17, 15 B. arithmetic; 2; 23, 25, 27 C. arithmetic; –3; 18, 15, 12 D. not arithmetic A B C D Example 1

Describe an Arithmetic Sequence Write an expression that can be used to find the nth term of the sequence 0.6, 1.2, 1.8, 2.4, …. Then write the next three terms of the sequence. Use a table to examine the sequence. The terms have a common difference of 0.6. Also, each term is 0.6 times its term number. Answer: An expression that can be used to find the nth term is 0.6n. The next three terms are 0.6(5) or 3, 0.6(6) or 3.6, and 0.6(7) or 4.2. Example 2

Write an expression that can be used to find the nth term of the sequence 1.5, 3, 4.5, 6, …. Then write the next three terms. A. 1.5n; 6.5, 7, 8.5 B. 1.5n; 7.5, 9, 10.5 C. 0.5n; 6.5, 7, 7.5 D. 2.5n; 8.5, 11, 13.5 A B C D Example 2

TRANSPORTATION This arithmetic sequence shows the cost of a taxi ride for 1, 2, 3, and 4 miles. What would be the cost of a 9-mile ride? +1 +1.75 The common difference between the costs is 1.75. This implies that the expression for the nth mile is 1.75n. Compare each cost to the value of 1.75n for each number of miles. Example 3

Each cost is 3. 50 more than 1. 75n. So, the expression 1. 75n + 3 Each cost is 3.50 more than 1.75n. So, the expression 1.75n + 3.50 is the cost of a taxi ride for n miles. To find the cost of a 9-mile ride, let c represent the cost. Then write and solve an equation for n = 9. Example 3

c = 1.75n + 3.50 Write the equation. c = 1.75(9) + 3.50 Replace n with 9. c = 15.75 + 3.50 or 19.25 Simplify. Answer: It would cost $19.25 for a 9-mile taxi ride. Example 3

TRANSPORTATION This arithmetic sequence shows the cost of a taxi ride for 1, 2, 3, and 4 miles. What would be the cost of a 15-mile ride? A B C D A. $18.75 B. $21.50 C. $24.50 D. $27.00 Example 3

Which expression can be used to find the nth term in the following arithmetic sequence, where n represents a number’s position in the sequence? A. n + 3 B. 3n C. 2n + 1 D. 3n – 1 Example 4

You need to find an expression to describe any term. Read the Item You need to find an expression to describe any term. Solve the Item The terms have a common difference of 3 for every increase in position number. So, the expression contains 3n. Eliminate choices A and C because they do not contain 3n. Eliminate choice B because 3(1) ≠ 2. The expression in choice D is correct for all the listed terms. So the correct answer is D. Answer: D Example 4

Let n represent the position of a number in the sequence 7, 11, 15, 19, … Which expression can be used to find any term in the sequence? A. 7n B. 4n – 3 C. 7 – n D. 4n + 3 A B C D Example 4

Solve 4m + 5 = 25. Then check your solution. (over Chapter 8) Solve 4m + 5 = 25. Then check your solution. A. B. C. D. A B C D Five Minute Check 1

(over Chapter 8) A. 27 B. 31 C. 35 D. 39 A B C D Five Minute Check 2

Solve the inequality –2x ≤ 5. Then check your solution. (over Chapter 8) Solve the inequality –2x ≤ 5. Then check your solution. A. B. C. D. A B C D Five Minute Check 3

Solve the inequality a + 2 > –7. Then check your solution. (over Chapter 8) Solve the inequality a + 2 > –7. Then check your solution. A. a < –9 B. a > –9 C. a > 9 D. a < 9 A B C D Five Minute Check 4

(over Chapter 8) Translate four less than five times a number is greater than nineteen into an inequality. A. 5x – 4 > 19 B. 5x – 4 < 19 C. 5x + 4 < 19 D. 5x + 4 > 19 A B C D Five Minute Check 5

(over Chapter 8) The perimeter of a square with side length x is no less than 36 inches. Which inequality represents all possible values for x? A. x > 9 B. x ≤ 4 C. x ≥ 9 D. x ≥ 4 A B C D Five Minute Check 6

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