Grade 10-Ppt-Week 1 (SEQUENCE and SERIES).pptx

lesson for grade 10, the sequence ad series

1. Good Day!

2. UNIT 1 1.1 SEQUENCE & SERIES PREPARED BY: GILLIAN F.REMOROZA Subject Teacher

3. Learning Objectives Explain and generate the concept of sequence. Solve the sequence problem Use formula to solve problems involving sequences, polynomials and polynomial equations.

4. Introduction Sequence and series are important concepts in many branches of mathematics. Their applications are numerous, particularly in biology, physics, investments, accountancy, and even in the field of arts. Nature is not a strange place for those mathematical concepts. Sequential patterns have been observed in beehive constructions, pine cone, sunflower and pineapple.

5. Introduction Sequence and series are important concepts in many branches of mathematics. Their applications are numerous, particularly in biology, physics, investments, accountancy, and even in the field of arts. Nature is not a strange place for those mathematical concepts. Sequential patterns have been observed in beehive constructions, pine cone, sunflower and pineapple.

6. Introduction Sequence and series are important concepts in many branches of mathematics. Their applications are numerous, particularly in biology, physics, investments, accountancy, and even in the field of arts. Nature is not a strange place for those mathematical concepts. Sequential patterns have been observed in beehive constructions, pine cone, sunflower and pineapple. Varied experiences with problem solving involving sequences and series as method of inquiry.

7. SEQUENCE A sequence is succession of numbers in a specific order. It can be finite or infinite TERM Each number in a sequence is called a term. The term are formed according to some fixed rule or property. They are arranged as the first term , the second term , the third term and so on. Examples: The sequence is infinite . The sequence is finite .   Example: In the sequence the first term is , the second term is , the third term is , and so on.  

8. SEQUENCE In the sequence the first and last terms of a sequence are referred to as extreme . The terms between the first and last term are called means . Example: Find the next three terms of each sequence. 25, 17, 9, …. 0.5, 1, 5, 4, 5, … 1, -4, 9, -16, ….. 4, 5, 9, 14, …. a. In the sequence 25, 17, 9… the next term is 8 less than any exceeding term. Hence, the next three terms are, 1, -7 and -15 .

9. SEQUENCE In the sequence the first and last terms of a sequence are referred to as extreme . The terms between the first and last term are called means . Example: Find the next three terms of each sequence. 25, 17, 9, …. 0.5, 1.5, 4.5, … 1, -4, 9, -16, ….. 4, 5, 9, 14, …. b. In the sequence 0.5, 1.5, 4.5, … a term is 3 times the preceding term. Thus, the next three terms are 13.5, 40.5, and 121.5 .

10. SEQUENCE In the sequence the first and last terms of a sequence are referred to as extreme . The terms between the first and last term are called means . Example: Find the next three terms of each sequence. 25, 17, 9, …. 0.5, 1.5, 4.5, … 1, -4, 9, -16, ….. 4, 5, 9, 14, …. c. In the sequence 1, -4, 9, -16,….. note that the terms have alternating signs, Moreover, the absolute value of each term corresponds to the square of a natural number. Therefore, the next three term are 25, -36 , and 49 .

11. SEQUENCE In the sequence the first and last terms of a sequence are referred to as extreme . The terms between the first and last term are called means . Example: Find the next three terms of each sequence. 25, 17, 9, …. 0.5, 1.5, 4.5, … 1, -4, 9, -16, ….. 4, 5, 9, 14, …. d. In the sequence 4, 5, 9, 14, … note that if then and . Therefore, fifth term is , the sixth term is , and the seventh term is  

12. Use the functional relation , where is a natural number, to write an infinite sequence.   A sequence is a function whose domain is the set of natural numbers or a subset of consecutive positive integers. Example: If   = 2-3 = -1  

13. Use the functional relation , where is a natural number, to write an infinite sequence.   A sequence is a function whose domain is the set of natural numbers or a subset of consecutive positive integers. Example: If   = 2-3 = -1  

14. Using the consecutive positive integers write the first five terms of the sequence defined by   Example: If     If    

15. Activity Time! Answer exercise 1.1 Mental Math, page 13.

16. You were born to win. But to be a winner, you must PLAN to win, PREPARE to win and EXPECT to win Aja! Fighting students.

17. Recursive Form of a Sequence A sequence can be expressed in recursive form . A sequence is said to be in recursive form if the first term and a recursive formula are given . A recursive formula is an expression used to determine the term of the sequence by using the term that precedes it.   Example: Find the next two terms in the given sequence, then write it in recursive form. 7, 12, 17, 22, 27…. 3, 7, 15, 31, 63…. a. Look for a pattern. Note that each term is 5 more than the preceding term. Label each term. Solution: 7 12 17 22 27          

18. Recursive Form of a Sequence A sequence can be expressed in recursive form . A sequence is said to be in recursive form if the first term and a recursive formula are given . A recursive formula is an expression used to determine the term of the sequence by using the term that precedes it.   Example: Find the next two terms in the given sequence, then write it in recursive form. 7, 12, 17, 22, 27…. 3, 7, 15, 31, 63…. Solution:                

19. Explicit Form of a Sequence A sequence can also be expressed in a form in which a preceding term is not necessary to find the succeeding terms. This explicit form can be used to find a term of the sequence by determining its position . Example: Determine the next two term in the given sequence. Then write the explicit form of the sequence. 1, 4, 9, 16, 25…. 3, 5, 9, 17, 33…. Solution: Let and . The subscript of are the natural number 1, 2, 3, 4, and 5, respectively. Notice that each term can be obtained by squaring its corresponding subscript. In tabular form,  

20. Explicit Form of a Sequence A sequence can also be expressed in a form in which a preceding term is not necessary to find the succeeding terms. This explicit form can be used to find a term of the sequence by determining its position . Example: Determine the next two term in the given sequence. Then write the explicit form of the sequence. 1, 4, 9, 16, 25…. 3, 5, 9, 17, 33…. Solution: 1 2 3 4 5 1 4 9 16 25 1 2 3 4 5 1 4 9 16 25 For the next two term and and  

21. Explicit Form of a Sequence A sequence can also be expressed in a form in which a preceding term is not necessary to find the succeeding terms. This explicit form can be used to find a term of the sequence by determining its position . Example: Determine the next two term in the given sequence. Then write the explicit form of the sequence. 1, 4, 9, 16, 25…. 3, 5, 9, 17, 33…. Solution: 1 2 3 4 5 1 4 9 16 25 1 2 3 4 5 1 4 9 16 25 For the next two term and and   The explicit form of the sequence is for every .  

22. SERIES

23. WHAT IS SERIES?

24. A SEQUENCE is defined as an arrangement of numbers in a particular order. On the other hand, a SERIES is defined as the SUM of the elements of a sequence .

25. SERIES The sum of the terms of the sequence , it can be expressed as . Series is denoted as , where refer to the number of terms.   SOLUTION:   Example: Find the indicated value for each series. a. 2 + 4 + 6 + 8….,  

26. SERIES The inconvenience of writing so many terms can be minimized using the summation notation. It makes use of the symbol (upper case sigma), a Greek letter equivalent to S which is the first letter of the word sum. The series 1 + 3+5 +7+9 can be written as, This read as “the  

27. Activity Time! Answer exercise 1.1 Written Math B, page 13. Number 11 to 16

28. You were born to win. But to be a winner, you must PLAN to win, PREPARE to win and EXPECT to win Aja! Fighting students.

29. Thank you for today’s class! .