ME Math 10 Q1 0401 PSHARMONIC SEQUENCE.pptx

HARMONIC SEQUENCE

1. Lesson 4 . 1 Harmonic Sequences

2. At the end of the lesson, the learners should be able to illustrate other types of sequences (e.g., harmonic, Fibonacci).

3. Properly identify a harmonic sequence. Correctly find the nth term of a harmonic sequence.

4. Correctly insert harmonic means between any two terms. Accurately solve word problems involving harmonic sequence.

5. What makes music beautiful and pleasing to the ears?

6. There are multiple reasons why we relate to a certain musical piece—it may be the lyrics, melody, rhythm, or harmony.

7. Why do you say that a certain chord progression is pleasing to the ears? In the Renaissance period, music theorists and composers created a tuning system which allowed integer ratios up to and including the integer 6.

8. In the later years, harmonic sequence was observed in natural phenomenon as well as in composition of beautiful music. In the case of musical harmonic sequence, the sequence has the form .  

9. How amazing it is that mathematics play a big role in music! Aside from that, these other types of sequences are greatly used and seen in our everyday lives. It is evident in the pattern of the stems in trees, speed of vehicles, or even in the measurement of raindrops. In this lesson, we will learn about harmonic sequence.

10. How can you identify a harmonic sequence? How can you find the nth term of a harmonic sequence? How can you insert harmonic mean/s between any two given terms of a harmonic sequence? 

11. This is a sequence whose terms are the reciprocals of the terms of an arithmetic sequence . Example: The sequence is harmonic because the reciprocal of its terms form the arithmetic sequence .   Harmonic Sequence

12. These are term(s) between any two nonconsecutive terms of a harmonic sequence. Example: In the harmonic sequence 1 and are harmonic means between and .   Harmonic Mean(s)

13. Example 1 : Is the sequence harmonic? Why?  

14. Solution : 1. Take the reciprocal of each term of the sequence.   Example 1 : Is the sequence harmonic? Why?  

15. Solution : 2. Check if the resulting sequence is arithmetic by determining if there is a common difference between consecutive terms.   Example 1 : Is the sequence harmonic? Why?  

16. Solution :   Example 1 : Is the sequence harmonic? Why?  

17. Solution : The sequence is arithmetic with a common difference of .   Example 1 : Is the sequence harmonic? Why?  

18. Solution : Therefore, the sequence is harmonic since the reciprocal of its terms form an arithmetic sequence.   Example 1 : Is the sequence harmonic? Why?  

19. Example 2 : What is the 30th term of the harmonic sequence ?  

20. Solution :   Take the reciprocal of each term of the sequence.     Example 2 : What is the 30th term of the harmonic sequence ?  

21. Solution :   2. Get the common difference by subtracting any term from its succeeding term. Let us subtract the first term from the second term. Example 2 : What is the 30th term of the harmonic sequence ?  

22. Solution :   Thus, the common difference is   Example 2 : What is the 30th term of the harmonic sequence ?  

23. Solution :   Solve the 30th term of the arithmetic sequence by using the formula for the nth term of an arithmetic sequence. We will use , where and .   Example 2 : What is the 30th term of the harmonic sequence ?  

24. Solution :     Example 2 : What is the 30th term of the harmonic sequence ?  

25. Solution :   Example 2 : What is the 30th term of the harmonic sequence ?  

26. Solution : Take the reciprocal of 21, which is .   Therefore, the 30th term of the harmonic sequence is .   Example 2 : What is the 30th term of the harmonic sequence ?  

27. Individual Practice: Which of the following sequences are harmonic? a. b. c. d.  

28. Individual Practice: Determine the position of the term in the harmonic sequence .  

29. Group Practice : To be done in groups of five. Two consecutive terms of a harmonic sequence are and . If the last term is and the 6th term is , how many terms are there in the sequence?  

30. A harmonic s equence is a sequence whose terms are reciprocal of the terms of an arithmetic sequence. Harmonic Mean(s) are term ( s) between any two nonconsecutive terms of a harmonic sequence.

31. “Harmonic Sequence.” Art of Problem Solving. Accessed June 2, 2022. https://artofproblemsolving.com/wiki/index.php/Harmonic_sequence# . “Harmonic Sequence.” Cuemath . Accessed June 2, 2022. https://www.cuemath.com/learn/mathematics/sequences-harmonic-sequence/ . Hosch , William. L. "harmonic sequence." Encyclopedia Britannica, May 25, 2016. https://www.britannica.com/science/harmonic-sequence-mathematics .