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1. Lesson 7.2 Series, Partial Sums, and Limiting Sums Precalculus Science, Technology, Engineering, and Mathematics

2. The population growth of a virus inside a body is exponential, which can be predicted at a certain time with the help of what we call a partial sum . 2

3. 3 In the previous lesson, we discussed sequences. In this lesson, we will learn about the sum of the terms of arithmetic and geometric sequences, which is called the partial sum . Moreover, in this lesson, we will be able to differentiate a sequence from a series.

4. 4 What is the difference between the partial sums and the infinite series?

5. 5 Illustrate a series (STEM-PC11SMI-Ih-1). Differentiate a series from a sequence (STEM-PC11SMI-Ih-2).

6. 6 Define a series. Differentiate partial sums and infinite series. Determine if an infinite series converges or diverges. Solve for partial sums and limiting sums.

7. 7 For the sequence   the partial sums are   Partial Sum of a Sequence

8. 8 is the first partial sum. is the second partial sum. is called the th partial sum.   Partial Sum of a Sequence

9. 9 The sequence   is called the sequence of partial sums .   Partial Sum of a Sequence

10. 10 Find the first four partial sums of the sequence  

11. 11   Find the first four partial sums of the sequence  

12. 12 12 F i nd the first five partial sums of the sequence .  

13. 13 Find the first four partial sums of the sequence whose th term is  

14. 14   Find the first four partial sums of the sequence whose th term is  

15. 15 15 Find the first four partial sums of the sequence whose th term is .  

16. 16 Find the first four partial sums and the th partial sum of the sequence whose th term is  

17. 17 or   Find the first four partial sums and the th partial sum of the sequence whose th term is  

18. 18 18 Find the first four partial sums and the th partial sum of the sequence whose th term is .  

19. 19 The partial sum for an arithmetic sequence , also known as the arithmetic series, refers to the sum of the terms of an arithmetic sequence. Partial Sum of an Arithmetic Sequence

20. 20 If and are given, the formula is given by   Partial Sum of an Arithmetic Sequence

21. 21 If and are given, the formula is given by   Partial Sum of an Arithmetic Sequence

22. 22 Example: Find the sum of the first 30 terms of the arithmetic sequence if the first term is 22 and the 18th term is 141. Partial Sum of an Arithmetic Sequence

23. 23 Example: Find the sum of the first 30 terms of the arithmetic sequence if the first term is 22 and the 18th term is 141. Partial Sum of an Arithmetic Sequence  

24. 24 The partial sum of a finite geometric sequence refers to the sum of the terms of a geometric sequence.   Partial Sum of a Finite Geometric Sequence

25. 25 The formula for a finite geometric series is given by where is the first term, and 𝑟 is the common ratio.   Partial Sum of a Finite Geometric Sequence

26. 26 Example: Find the sum of the first 6 terms of a geometric sequence whose first term is 8, and the common ratio is 3. Partial Sum of a Finite Geometric Sequence

27. 27 Example: Find the sum of the first 6 terms of a geometric sequence whose first term is 8, and the common ratio is 3. Partial Sum of a Finite Geometric Sequence  

28. 28 Find the sum of the first 50 terms of the arithmetic sequence if the first term is 20 and the 15th term is 132.

29. 29 10 800 Find the sum of the first 50 terms of the arithmetic sequence if the first term is 20 and the 15th term is 132.

30. 30 30 Find the sum of the first 8 terms of the geometric sequence whose first term is 4, and the common ratio is .  

31. 31 A pyramid of cups has 40 pieces in the bottom row and one fewer cup in each successive row. How many cups are there in the pyramid?

32. 32 820 cups A pyramid of cups has 40 pieces in the bottom row and one fewer cup in each successive row. How many cups are there in the pyramid?

33. 33 33 A movie theater has 30 rows of seats. In the first row, there are 25 seats. Additional two seats are placed for each succeeding row. How many seats are there in the theater?

34. 34 Justine saved ₱20 in the first week of school. Suppose in any succeeding week, he always saves twice the amount he saved in the previous week. How much is his total savings after 3 months?

35. 35 ₱81 900 Justine saved ₱20 in the first week of school. Suppose in any succeeding week, he always saves twice the amount he saved in the previous week. How much is his total savings after 3 months?

36. 36 36 Joana started collecting marbles. In the first week, she has 3 marbles. She doubles her collection every week. How many marbles will she have after 1 and a half months?

37. 37 In a sequence, we have both finite and infinite sequences. Are partial sums infinite or finite?

38. 38 Note that we are only getting the sum of a part of a sequence. Since we can only have a certain number of parts that we want to consider, a partial sum is finite. That is why partial sums are also sometimes called a finite series .

39. 39 An infinite series is the sum of all the terms in an infinite sequence. Infinite Series

40. 40 Infinite Convergent Series and Infinite Divergent Series When adding the first few terms of a sequence, and it approaches a finite value 𝑆, we can say that an infinite series converges (or is convergent) ; otherwise, it diverges (or is divergent) . Infinite Series

41. 41 Example: The series is convergent. The series is divergent.   Infinite Series

42. 42 The limiting sum of an infinite geometric series is for   Limiting Sum of Infinite Geometric Series

43. 43 Example: Find the limiting sum of the infinite geometric series if it exists.   Limiting Sum of Infinite Geometric Series

44. 44 Example: Find the limiting sum of the infinite geometric series if it exists.   Limiting Sum of Infinite Geometric Series  

45. 45 Determine whether the infinite series converges or diverges.  

46. 46 The series is divergent. Determine whether the infinite series converges or diverges.  

47. 47 47 Determine whether the infinite series converges or diverges.  

48. 48 Find the limiting sum of the infinite geometric series if it exists.  

49. 49   Find the limiting sum of the infinite geometric series if it exists.  

50. 50 50 Find the limiting sum of the infinite geometric series if it exists.  

51. 51 Mark is playing basketball with his friends. In the last few seconds of the game, Mark shoots the ball from the three-point line and scored, which made their team win the game. As the ball bounced through the ring, it falls to the ground and rebounds half the height, then it comes to a rest, and rolls on the ground. Solve for the total vertical distance traveled by the ball if the ring is 10 feet high.

52. 52 30 feet Mark is playing basketball with his friends. In the last few seconds of the game, Mark shoots the ball from the three-point line and scored, which made their team win the game. As the ball bounced through the ring, it falls to the ground and rebounds half the height, then it comes to a rest, and rolls on the ground. Solve for the total vertical distance traveled by the ball if the ring is 10 feet high.

53. 53 53 The antibacterial property of a plant is to be tested on a petri dish with a bacteria population of 10 000 cfu /ml (colony forming units per milliliter). When injected to an agar of the petri dish, it reduced the population of the bacteria by a quarter of its original population. Solve for the limiting sum of the infinite geometric series.

54. 54 In what condition can we solve for the limiting sum of an infinite geometric series?

55. 55 Given the following sequence, provide its first four partial sums and th term of the partial sums.   Sequence First Four Partial Sums t h term of the Partial Sums 1. 2. Sequence First Four Partial Sums

56. 56 Solve for the partial sum of the arithmetic/geometric sequence below. 1. arithmetic sequence: 2 . geometric sequence: 3. Find the sum of the first 10 terms of the geometric sequence whose first term is -8 and  

57. 57 A series refers to the sum of the terms in a sequence. Partial sum is the sum of a part of a sequence.

58. 58 Infinite series is the sum of all the terms in an infinite sequence. An infinite series is convergent if it approaches a specific value. Otherwise, it is divergent .

59. 59 Concept Formula Description Partial Sum of an Arithmetic Sequence where is the partial sum; is the number of terms; is the first term, and is the nth term . Use this formula when the first and last term of the sequence are given. Concept Formula Description Partial Sum of an Arithmetic Sequence Use this formula when the first and last term of the sequence are given.

60. 60 Concept Formula Description Partial Sum of an Arithmetic Sequence where is the partial sum; is the number of terms; is the first term, and is the common difference . Use this formula when the first term and the common difference are given. Concept Formula Description Partial Sum of an Arithmetic Sequence Use this formula when the first term and the common difference are given.

61. 61 Concept Formula Description Partial Sum of a Geometric Sequence where is the partial sum; is the number of terms; is the first term, and is the ratio . Use this formula to solve for the partial sum of a finite geometric sequence . Concept Formula Description Partial Sum of a Geometric Sequence Use this formula to solve for the partial sum of a finite geometric sequence .

62. 62 Concept Formula Description Limiting Sum/ Sum of Infinite Geometric Sequence where is the partial sum; is the first term, and is the ratio. Use this formula to solve for the limiting sum of an infinite geometric sequence . Concept Formula Description Limiting Sum/ Sum of Infinite Geometric Sequence Use this formula to solve for the limiting sum of an infinite geometric sequence .

63. 63 63 Ynnah’s annual salary in her work is ₱85 000. If she receives an increase of 25% yearly, how much will be her annual salary after 5 years?

64. 64 Barnett, Raymond, Michael Ziegler, Karl Byleen , and David Sobecki . College Algebra with Trigonometry. Boston: McGraw Hill Higher Education, 2008. Bittinger , Marvin L., Judith A. Beecher, David J. Ellenbogen , and Judith A. Penna. Algebra and Trigonometry: Graphs and Models. 4th ed. Boston: Pearson/Addison Wesley, 2009. Blitzer, Robert. Algebra and Trigonometry. 3rd ed. Upper Saddle River, New Jersey: Pearson/Prentice Hal, 2007. Larson, Ron. College Algebra with Applications for Business and the Life Sciences. Boston: MA: Houghton Mifflin, 2009. Simmons, George F. Calculus with Analytic Geometry. 2nd ed. New York: McGraw-Hill, 1996.