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Concept

A. Determine whether the sequence is arithmetic. –3, –8, –13, –23, … Identify Arithmetic Sequences A. Determine whether the sequence is arithmetic. –3, –8, –13, –23, … –5 –10 –3 –8 –13 –23 There is no common difference. Answer: This is not an arithmetic sequence. Example 1A

B. Determine whether the sequence is arithmetic. –8, –2, 4, 10, … Identify Arithmetic Sequences B. Determine whether the sequence is arithmetic. –8, –2, 4, 10, … +6 –8 –2 4 10 The common difference is 6. Answer: The sequence is arithmetic. Example 1B

A. The common difference is 9. B. The common difference is 11. A. Determine whether the sequence is arithmetic. If so, determine the common difference. –16, –5, 6, 17, … A. The common difference is 9. B. The common difference is 11. C. The common difference is 13. D. The sequence is not arithmetic. Example 1A

Answer: The next four terms are –2, 0, 2, and 4. Graph an Arithmetic Sequence A. Consider the arithmetic sequence –8, –6, –4, …. Find the next four terms of the sequence. Step 1 To determine the common difference, subtract any term from the term directly after it. The common difference is –4 – (–6) or 2. Step 2 To find the next term, add 2 to the last term. Continue to add 2 to find the following terms. +2 –8 –6 –4 –2 0 2 4 Answer: The next four terms are –2, 0, 2, and 4. Example 2A

Graph an Arithmetic Sequence B. Consider the arithmetic sequence –8, –6, –4, …. Graph the first seven terms of the sequence. The domain contains the terms {1, 2, 3, 4, 5, 6, 7} and the range contains the terms {–8, –6, –4, –2, 0, 2, 4}. So, graph the corresponding pairs. Answer: Example 2B

A. Consider the arithmetic sequence 22, 13, 4, … A. Consider the arithmetic sequence 22, 13, 4, … . Find the next four terms of the sequence. A. 4, 8, 12, and 16 B. 6, 15, 24, and 33 C. 3, 12, 21, and 30 D. 5, 14, 23, and 32 Example 2A

B. Consider the arithmetic sequence 22, 13, 4, … B. Consider the arithmetic sequence 22, 13, 4, … . Graph the first seven terms of the sequence. B. C. D. Example 2B

Find a Term MARCHING BAND During their routine, a high school marching band marches in rows. There is one performer in the first row, three performers in the next row, and five in the third row. This pattern continues for the rest of the rows. Use this information to determine how many performers will be in the 20th row during the routine. Understand Because the difference between any two consecutive rows is 2, the common difference for the sequence is 2. Plan Use point-slope form to write an equation for the sequence. Let m = 2 and (x1, y1) = (3, 5). Then solve for x = 20. Example 3

Solve (y – y1) = m(x – x1) Point-slope form Find a Term Solve (y – y1) = m(x – x1) Point-slope form (y – 5) = 2(x – 3) m = 2 and (x1, y1) = (3, 5) y – 5 = 2x – 6 Multiply. y = 2x – 1 Add 5 to each side. y = 2(20) – 1 Replace x with 20. y = 40 – 1 or 39 Simplify. Example 3

Answer: There will be 39 performers in the 20th row. Find a Term Answer: There will be 39 performers in the 20th row. Check You can find the terms of the sequence by adding 2, starting with row 1, until you reach 20. Example 3

PYRAMIDS Hermán is building a pyramid out of blocks for an engineering class. On the top level, there is one block. In the second level, there are 5 blocks. In the third, there are 9 blocks. This pattern continues for the rest of the levels down to the 18th level at the base of the pyramid. Use this information to determine how many blocks will be in the 13th level of the pyramid. A. 41 blocks B. 45 blocks C. 49 blocks D. 53 blocks Example 3

A. Determine whether the sequence is geometric. 8, 20, 50, 125, … Identify Geometric Sequences A. Determine whether the sequence is geometric. 8, 20, 50, 125, … Find the ratios of the consecutive terms. Answer: The ratios are the same, so the sequence is geometric. Example 4A

B. Determine whether the sequence is geometric. 19, 30, 41, 52, … Identify Geometric Sequences B. Determine whether the sequence is geometric. 19, 30, 41, 52, … Answer: The ratios are not the same, so the sequence is not geometric. Example 4B

A B A. Determine whether the sequence is geometric. –4, 8, –16, 32, … A. The sequence is geometric. B. The sequence is not geometric. A B Example 4A

Step 1 Find the value of the common ratio: Graph a Geometric Sequence A. Consider the geometric sequence 10, 15, 22.5, … . Find the next three terms of the sequence. Step 1 Find the value of the common ratio: Step 2 To find the next term, multiply the previous term by Continue multiplying by to find the following terms. Example 5A

Answer: The next three terms are 33.75, 50.625, and 75.938. Graph a Geometric Sequence 10 15 22.5 33.75 50.625 75.938 Answer: The next three terms are 33.75, 50.625, and 75.938. Example 5A

B. Graph the first six terms of the sequence. Graph a Geometric Sequence B. Graph the first six terms of the sequence. Domain: {1, 2, 3, 4, 5, and 6} Range: {10, 15, 22.5, 33.75, 50.625, 75.938} Answer: Example 5B

A. Consider the geometric sequence 2, 6, 18, … A. Consider the geometric sequence 2, 6, 18, … . Find the next three terms of the sequence. A. 30, 42, 54 B. 36, 72, 144 C. 72, 288, 1152 D. 54, 162, 486 Example 5A

B. Graph the first six terms of the sequence. C. D. Example 5B

Check for a common difference. 49 – 37 = 12 37 – 25 = 12 25 – 13 = 12 Classify Sequences A. Determine whether the sequence is arithmetic, geometric, or neither. Explain your reasoning. 13, 25, 37, 49, … Check for a common difference. 49 – 37 = 12 37 – 25 = 12 25 – 13 = 12  Check for a common ratio.  Answer: Because there is a common difference, the sequence is arithmetic. Example 6A

Check for a common difference. 14 – 9 = 5 9 – 5 = 4  Classify Sequences B. Determine whether the sequence is arithmetic, geometric, or neither. Explain your reasoning. 2, 5, 9, 14, … Check for a common difference. 14 – 9 = 5 9 – 5 = 4  Check for a common ratio.  Answer: Because there is no common difference or common ratio, the sequence is neither arithmetic nor geometric. Example 6B

Check for a common difference. –48 – 24 = –72 24 – (–12) = 36  Classify Sequences C. Determine whether the sequence is arithmetic, geometric, or neither. Explain your reasoning. 6, –12, 24, –48, … Check for a common difference. –48 – 24 = –72 24 – (–12) = 36  Check for a common ratio.  Answer: Because there is a common ratio, the sequence is geometric. Example 6C

A. The sequence is arithmetic. B. The sequence is geometric. A. Determine whether the sequence is arithmetic, geometric, or neither. Explain your reasoning. 8, 24, 48, 96, … A. The sequence is arithmetic. B. The sequence is geometric. C. The sequence is neither. Example 6A

A. The sequence is arithmetic. B. The sequence is geometric. B. Determine whether the sequence is arithmetic, geometric, or neither. Explain your reasoning. 5, 12, 19, 26, … A. The sequence is arithmetic. B. The sequence is geometric. C. The sequence is neither. Example 6B

A. The sequence is arithmetic. B. The sequence is geometric. C. Determine whether the sequence is arithmetic, geometric, or neither. Explain your reasoning. 300, 200, … A. The sequence is arithmetic. B. The sequence is geometric. C. The sequence is neither. Example 6C

Homework P. 663 # 1 – 7 odd, 12 – 48 (x3)

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