Patterns & Arithmetic Sequences.pptx

These are numbers, shapes, or objects that are arranged according to a rule.

1. Patterns and Sequences

2. At the end of the lessons, the learner should be able to do the following: Objectives : Generate and illustrate patterns. Identify the next term in a given pattern. Find the general rule for the th term of a sequence.  

3. We are all surrounded by different patterns. From the swirling pattern of a nautilus shell to the geometric pattern of honeycombs, we can observe several mathematical patterns that define the beauty of nature.

4. In fact, it was the Greeks who used a particular ratio, called the golden ratio , to design many of their architectural structures, such as the Parthenon. They believe that such pattern is aesthetically pleasing and easy on the eyes.

5. In this lesson, we will learn more about several sequences and series that make up mathematical patterns.

6. These are numbers, shapes, or objects that are arranged according to a rule. Example: Patterns

7. A sequence is a set of values that follow a particular order. It can be finite or infinite . Examples: The sequence is infinite. The sequence is finite.   Sequence

8. Each value in a sequence is called a term . Example: In the sequence the first term is , the second term is , the third term is , and so on.   Term

9. Each value in a sequence is called a term . Example: The th term of the sequence is where is a positive integer. The expression is called the general term of the sequence.   Term

10. The sum of the terms in a sequence is called a series . Examples: 1. 2.   Series

11. Example 1: Find the first five terms of the sequence whose th term is .  

12. Solution: The domain of the sequence is . Substituting the values in the given expression will give the following terms on the next slide.   Example 1: Find the first five terms of the sequence whose th term is .  

13. Solution:   Example 1: Find the first five terms of the sequence whose th term is .  

14. Solution: Therefore, the first five terms of the sequence are 2 , 7 , 12 , 17 , and 22 . Example 1: Find the first five terms of the sequence whose th term is .  

15. Example 2: Find the eighth term of the sequence whose th term is .  

16. Solution: We will substitute 8 for since we are looking for the eighth term.   Example 2: Find the eighth term of the sequence whose th term is .  

17. Solution: Therefore, the 8th term is 12 .   Example 2: Find the eighth term of the sequence whose th term is .  

18. Lesson 1 . 2 Arithmetic Sequences

19. Did you know that you can get the approximate age (in years) of a tree based on its number of rings? Fig. 1. Tree Rings

20. The number of rings indicate the annual pattern of the growth of a tree. This pattern is an example of a sequence .

21. In this lesson, we will study a sequence that exhibits the same pattern with the tree rings.

22. An arithmetic sequence is a sequence where each term after the first is obtained by adding a constant (called the common difference ) to the preceding term. Arithmetic Sequence Example:  

23. A term refers to each value in a sequence. Term Example: The 1st term is 4, 2nd term is 8, 3rd term is 12, 4th term is 16 and so on.  

24. The general term of a sequence is the formula used to find the th term of a sequence, which is , where is the th term, is the first term, is the number of terms, and is the common difference.   General Term of a Sequence

25. Example: In the sequence , the 10th term can be found by substituting the values to the general term formula.   General Term of a Sequence

26. Example 1 : Find the 20th term in the sequence  

27. Find the 20th term in the sequence Solution : I dentify the common difference of the sequence.   Thus, the common difference is 7.    

28. Find the 20th term in the sequence Solution : List all information needed to use the formula for the general term.  

29. Find the 20th term in the sequence Solution : Substitute the given information to the formula.    

30. Find the 20th term in the sequence Solution :   Therefore, the 20th term of the given sequence is 139 .  

31. Example 2 : Find the 30th term of an arithmetic sequence if the 5th term is and the common difference is 6.  

32. Find the 30th term of an arithmetic sequence if the 5th term is and the common difference is 6. Solution : Use the formula for arithmetic sequence, then substitute the given information. Solve the first term by substituting , , and to the formula.  

33. Find the 30th term of an arithmetic sequence if the 5th term is and the common difference is 6. Solution :    

34. Find the 30th term of an arithmetic sequence if the 5th term is and the common difference is 6. Solution : Using , solve for the 30th term of the sequence.    

35. Find the 30th term of an arithmetic sequence if the 5th term is and the common difference is 6. Solution : Therefore, the 30th term of the sequence is 140 .  

36. Individual Practice: Write the first eight terms of a sequence with the general formula . Find the 20th term of the sequence whose th term is given by . Given the arithmetic sequence , find the 10th term. 4. The 4th term of an arithmetic sequence is and the 6th term is . Write the first eight terms of the sequence.  

37. Group Practice: 1. The grade 10 students formed a human pyramid in the school field demonstration. The first row had one student and the 3rd row had 5 students. If there were 8 rows, how many students were there in the last row? 2. In January 2010, a new car was valued at ₱ 800 000. From then on, it depreciated in value by a regular amount each year. By January 2019, its market value is only ₱ 102 500. Show that the value of the car could follow an arithmetic sequence. How many terms are there in this sequence? By how much did it depreciate each year?

38. Patterns are numbers, shapes, or objects that are arranged according to a rule. A sequence is a set of values that follow a particular order. It can be finite or infinite . A term refers to each value in a sequence. A series is the sum of the terms in a sequence. Key Points:

39. Arithmetic Sequence An arithmetic sequence is a sequence where each term after the first is obtained by adding a constant (called the common difference) to the preceding term. Term A term refers to each value in the sequence. General Term of a Sequence The general term of a sequence refers to the formula used to find the th term of a sequence, which is , where is the th term, is the first term, is the number of terms and is the common difference.   Key Points: