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HARMONIC PROGRESSION

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  1. HARMONIC PROGRESSION

  2. What is a Harmonic Progression?

  3. A Harmonic Progression is a sequence of quantities whose reciprocals form an arithmetic progression.

  4. Note!

  5. * The series formed by the reciprocals of the terms of a geometric series are also geometric series.

  6. And

  7. * There is no general method of finding the sum of a harmonic progression. 

  8. Example

  9. The Sequence“s1 , s2 , … , sn”is a Harmonic Progression if “1/s1 , 1/s2 , … , 1/sn”forms an Arithmetic Progression.

  10. Method For Re-checking a Harmonic Progression

  11. A Harmonic Progression is a set of values that, once reciprocated, results to an Arithmetic Progression. To check , the reciprocated values must possess a rational common difference. Once this has been identified, we may say that the sequence is a Harmonic Progression.

  12. Harmonic Means are the terms found in between two terms of a harmonic progression.

  13. Problems

  14. Determine which of the following are Harmonic Progressions.

  15. 1 ,1/2 , 1/3 , 1/4 , ...

  16. Step 1: Reciprocate all the given terms.* The reciprocals are: 1 , 2 , 3 , 4 , …Step 2: Identify whether the reciprocated sequence is an Arithmetic Progression by checking if a common difference exists in the terms.

  17. Answer: It is a Harmonic Progression.

  18. 2) 1 , 1/4 , 1/5 , 1/7 , ...

  19. Step 1: Reciprocate all the given terms.* The reciprocals are: 1 , 4 , 5 , 7 , …Step 2: Identify whether the reciprocated sequence is an Arithmetic Progression by checking if a common difference exists in the terms.

  20. Answer: It is NOT a Harmonic Progression.

  21. Determine the next three terms of each of the following Harmonic Progressions.

  22. 1) 24 , 12 , 8 , 6 , …

  23. Solution: 24 , 12 , 8 , 6 , … = 1/24 , 1/12 , 1/8 , 1/6* To find the common difference: 1/12 – 1/24 = 2/24 – 1/24 = 1/24

  24. Note!

  25. You can subtract the second term to the first term, the third to the second term, the forth to the third term, and so on and so forth.

  26. To get the next three terms: 5th Term = 1/6 + 1/24 = 4/24 + 1/24 = 5/24* Reciprocate = 24/5

  27. 6th Term = 5/24 + 1/24 = 6/24 = 1/4* Reciprocate = 4

  28. 7th Term = 1/4 + 1/24 = 6/24 + 1/24 = 7/24* Reciprocate = 24/7

  29. Find the Harmonic Mean between the following terms.

  30. 1) 12 and 8

  31. Step 1: Reciprocate all the given terms.* The reciprocals are: 1/12 and 1/8Step 2: Arrange the given terms as follows:

  32. 1/12Harmonic Mean1/81’st term2’nd term3’rd term

  33. *For this problem, we will use the formula:tn = t1 + (n – 1)d

  34. We may now substitute the values in the problem to the formula to find the common difference (d) and the Harmonic Mean as follows:

  35. t3 = t1 + (3 - 1)d1/8 = 1/12 + 2d1/8 – 1/12 = 2d(3 – 2) / 24 = 2d(3 – 2) = 48d1 = 48dd = 1/48

  36. *After getting the Common Difference, add it to the first term to get the Harmonic Mean between the two terms.

  37. t2 = t1 + d= 1/12 + 1/48= (4 + 1) / 48= 5/48*Reciprocate= 48/5

  38. Insert three Harmonic Means between the following terms:

  39. 1) 36 and 36/5

  40. Step 1: Reciprocate all the given terms.* The reciprocals are: 1/36 and 5/36Step 2: Arrange the given terms as follows:

  41. 1/361’st termHarmonic Means2’nd , 3’rd , and 4’th term5/365’th term

  42. *For this problem, we will use the formula:tn = t1 + (n – 1)d

  43. We may now substitute the values in the problem to the formula to find the common difference (d) and the Harmonic Means as follows:

  44. t5 = t1 + (5 - 1)d5/36 = 1/36 + 4d5/36 – 1/36 = 4d(5 - 1) / 36 = 4d(5 - 1) = 144d4 = 144dd = 4/144 = 1/36

  45. *After getting the Common Difference, add it to the first term, then add it to the second term, and then add it to the third term to get the Harmonic Means between the two terms.

  46. t2 = t1 + d= 1/36 + 1/36= 2/36= 1/18*Reciprocate= 18

  47. t3 = t2 + d= 2/36 + 1/36= 3/36= 1/12*Reciprocate= 12

  48. t4 = t3 + d= 3/36 + 1/36= 4/36= 1/9*Reciprocate= 9

  49. Therefore, the three means between 36 and 36/5 are 18, 12, and 9.

  50. Activity