1. Splash Screen

2. Lesson Menu 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

3. Over Lesson 10–3 5-Minute Check 1 Find the first five terms of the geometric sequence for which a 1 = 625 and r =. __ 1 5 A.625, 25, 5, 1, B.625, 125, 25, 5, 1 C.625, 125, 5,, D.625, 125, 25, 5, __ 1 5 1 5 1 25 __ 1 5

4. Over Lesson 10–3 5-Minute Check 2 A.9 B.10 C.11 D.12 Find the sixth term of the geometric sequence for which a 1 = 352 and r =. __ 1 2

5. Over Lesson 10–3 5-Minute Check 3 Write an equation for the nth term of the geometric sequence 216, –36, 6, …. A. B.a n = 216(–6) n C.a n = 216(–36) n –1 D.a n = 216 + (n – 1)252

6. Over Lesson 10–3 5-Minute Check 4 A.5125 B.5120 C.5115 D.5110

7. Over Lesson 10–3 5-Minute Check 5 A.$18,240 B.$14,592 C.$7471.10 D.$4560 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?

8. Over Lesson 10–3 5-Minute Check 6 A.18 or –18 B.27 or –27 C.36 or –36 D.45 or –45 Find the geometric mean between 9 and 81.

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

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

11. Vocabulary infinite geometric series convergent series divergent series infinity

12. Concept

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

14. Example 1B 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.

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

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

17. Concept

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

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

20. Example 2B Sum of an Infinite Series Simplify. Answer: The sum of the series is 2. a 1 = 3, r =

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

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

23. Example 3 Infinite Series in Sigma Notation Sum formula Simplify. a 1 = 5, r = Evaluate. Answer: Thus,

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

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

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

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

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

29. End of the Lesson