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Splash Screen. Five-Minute Check (over Lesson 10–3) CCSS Then/Now New Vocabulary Key Concept: Convergent and Divergent Series Example 1: Convergent and Divergent Series Key Concept: Sum of an Infinite Geometric Series Example 2: Sum of an Infinite Series

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 10–3) CCSS Then/Now New Vocabulary Key Concept: Convergent and Divergent Series Example 1: Convergent and Divergent Series Key Concept: Sum of an Infinite Geometric Series Example 2: Sum of an Infinite Series Example 3: Infinite Series in Sigma Notation Example 4: Write a Repeating Decimal as a Fraction Lesson Menu

  3. Find the first five terms of the geometric sequence for which a1 = 625 and r = . A. 625, 25, 5, 1, B. 625, 125, 25, 5, 1 C. 625, 125, 5, , D. 625, 125, 25, 5, 1 1 1 1 1 __ __ __ __ __ 5 5 25 5 5 5-Minute Check 1

  4. Find the sixth term of the geometric sequence for which a1 = 352 and r = . 1 __ 2 A. 9 B. 10 C. 11 D. 12 5-Minute Check 2

  5. A. B.an = 216(–6)n C.an = 216(–36)n –1 D.an = 216 + (n – 1)252 Write an equation for the nth term of the geometric sequence 216, –36, 6, … . 5-Minute Check 3

  6. A. 5125 B. 5120 C. 5115 D. 5110 5-Minute Check 4

  7. A certain model automobile depreciates 20% of its value each year. If it costs $22,800 new, what is its value at the end of 5 years? A. $18,240 B. $14,592 C. $7471.10 D. $4560 5-Minute Check 5

  8. Find the geometric mean between 9 and 81. A. 18 or –18 B. 27 or –27 C. 36 or –36 D. 45 or –45 5-Minute Check 6

  9. Mathematical Practices 6 Attend to precision. 8 Look for and express regularity in repeated reasoning. CCSS

  10. You found sums of finite geometric series. • Find sums of infinite geometric series. • Write repeating decimals as fractions. Then/Now

  11. infinite geometric series • convergent series • divergent series • infinity Vocabulary

  12. Concept

  13. Answer: Since the series is convergent. Convergent and Divergent Series A. Determine whether the infinite geometric series is convergent or divergent.729 + 243 + 81 + … Find the value of r. Example 1A

  14. Convergent and Divergent Series B. Determine whether the infinite geometric series is convergent or divergent.2 + 5 + 12.5 + … Answer: Since 2.5 > 1, the series is divergent. Example 1B

  15. A. Determine whether the infinite geometric series is convergent or divergent.343 + 49 + 7 + … A. convergent B. divergent Example 1A

  16. B. Determine whether the infinite geometric series is convergent or divergent.4 + 14 + 49 + … A. convergent B. divergent Example 1B

  17. Concept

  18. A. Find the sum of , if it exists. the series diverges and the sum does not exist. Sum of an Infinite Series Find the value of r to determine if the sum exists. Answer: The sum does not exist. Example 2A

  19. B. Find the sum of , if it exists. the sum exists. Sum of an Infinite Series Now use the formula for the sum of an infinite geometric series. Sum formula Example 2B

  20. a1 = 3, r = Sum of an Infinite Series Simplify. Answer: The sum of the series is 2. Example 2B

  21. A. Find the sum of the infinite geometric series, if it exists.2 + 4 + 8 + 16 + ... A. 4 B. 1 C. 2 D. no sum Example 2A

  22. B. Find the sum of the infinite geometric series, if it exists. A. 4 B. 2 C. 1 D. no sum Example 2

  23. Evaluate . a1 = 5, r = Answer: Thus, Infinite Series in Sigma Notation Sum formula Simplify. Example 3

  24. Evaluate . A.6 B.3 C. D.no sum Example 3

  25. Write 0.25 as a fraction. Write a Repeating Decimal as a Fraction Method 1 Use the sum of an infinite series. Write the repeating decimal as a sum. Sum formula Example 4

  26. Write a Repeating Decimal as a Fraction Subtract. Simplify. Example 4

  27. Subtract S from 100Sand 0.25 from 25.25. Answer: Thus, . Write a Repeating Decimal as a Fraction Method 2 Use algebraic properties. Label the given decimal. Write as a repeating decimal. Multiply each side by 100. Divide each side by 99. Example 4

  28. Write 0.37 as a fraction. A. B. C. D. Example 4

  29. End of the Lesson

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