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Geometric Sequences

12.1. Geometric Sequences. Warm Up. Lesson Presentation. Lesson Quiz. Holt McDougal Algebra 1. Holt Algebra 1. 12.1. Warm Up Find the value of each expression. 1. 2 5. 32. 2. 2 –5. 3. –3 4. –81. 4. (–3) 4. 81. 5. (0.2) 3. 6. 7(–4) 2. 0.008. 112. 7. 8. 12(–0.4) 3.

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Geometric Sequences

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  1. 12.1 Geometric Sequences Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 1 Holt Algebra 1

  2. 12.1 • Warm Up • Find the value of each expression. 1. 25 32 2. 2–5 3. –34 –81 4. (–3)4 81 5. (0.2)3 6. 7(–4)2 0.008 112 7. 8. 12(–0.4)3 –0.768

  3. 12.1 The table shows the heights of a bungee jumper’s bounces. The height of the bounces shown in the table above form a geometric sequence. In a geometric sequence, the ratio of successive terms is the same number r, called the common ratio.

  4. 1 2 3 4 Position 3 6 12 24 Term 12.1 Geometric sequences can be thought of as functions. The term number, or position in the sequence, is the input, and the term itself is the output. a1a2a3a4 To find a term in a geometric sequence, multiply the previous term by r.

  5. 12.1 Example 1: Extending Geometric Sequences Find the next three terms in the geometric sequence. 1, 4, 16, 64,… Step 1 Find the value of r by dividing each term by the one before it. 1 4 16 64 The value of r is 4.

  6. 4 4 4 12.1 Example 1 Continued Find the next three terms in the geometric sequence. 1, 4, 16, 64,… Step 2 Multiply each term by 4 to find the next three terms. 64 256 1024 4096 The next three terms are 256, 1024, and 4096.

  7. 12.1 Example 2 Find the next three terms in the geometric sequence. 512, 384, 288,… Step 1 Find the value of r by dividing each term by the one before it. 512 384 288 The value of r is 0.75.

  8. 12.1 Example 2 Continued Find the next three terms in the geometric sequence. 512, 384, 288,… Step 2 Multiply each term by 0.75 to find the next three terms. 288 216 162 121.5 0.75 0.75 0.75 The next three terms are 216, 162, and 121.5.

  9. nth term 1st term Common ratio 12.1 If the first term of a geometric sequence is a1, the nth term is an , and the common ratio is r, then an=a1rn–1

  10. 12.1 Example 3: Finding the nth Term of a Geometric Sequence The first term of a geometric sequence is 500, and the common ratio is 0.2. What is the 7th term of the sequence? an = a1rn–1 Write the formula. a7 = 500(0.2)7–1 Substitute 500 for a1,7 for n, and 0.2 for r. Simplify the exponent. = 500(0.2)6 Use a calculator. = 0.032 The 7th term of the sequence is 0.032.

  11. 2 –6 18 –54 12.1 Example 4: Finding the nth Term of a Geometric Sequence What is the 9th term of the geometric sequence 2, –6, 18, –54, …? The value of r is –3. an = a1rn–1 Write the formula. Substitute 2 for a1,9 for n, and –3 for r. a9 = 2(–3)9–1 = 2(–3)8 Simplify the exponent. Use a calculator. = 13,122 The 9th term of the sequence is 13,122.

  12. Caution When writing a function rule for a sequence with a negative common ratio, remember to enclose r in parentheses. –212 ≠ (–2)12 12.1

  13. 12.1 Example 5 The table shows a car’s value for 3 years after it is purchased. The values form a geometric sequence. How much will the car be worth in the 10th year? 10,000 8,000 6,400 The value of r is 0.8.

  14. 12.1 Example 5 an = a1rn–1 Write the formula. Substitute 10,000 for a1,10 for n, and 0.8 for r. a6 = 10,000(0.8)10–1 = 10,000(0.8)9 Simplify the exponent. Use a calculator. = 1,342.18 In the 10th year, the car will be worth $1342.18.

  15. 12.1 Example 6 Given the geometric sequence below, find the common ratio and the explicit formula. 5, -10, 20, -40, … an = a1rn–1 Write the formula. Substitute 5 for a1and -2 for r. an= 5(-2)n–1 You CANNOT simplify.

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