Week 2.1 Sequence and Series.pdf

CLASSROOM RULES: H onesty Always be truthful and act with integrity. Avoid cheating, lying, or plagiarizing. A ttentiveness Pay close attention to the teacher and classmates....

CLASSROOM RULES: H onesty Always be truthful and act with integrity. Avoid cheating, lying, or plagiarizing. A ttentiveness Pay close attention to the teacher and classmates. Listen actively and stay focused during lessons and discussions. S afety Follow all safety protocols and guidelines. Ensure the classroom is a safe environment for everyone. Treat classmates, teachers, and school property with M utual respect respect. Use polite language and be considerate of others' opinions and feelings. I nvolvement Participate actively in class activities and discussions. Engage in group work and contribute your ideas. N eatness Keep your workspace and the classroom clean and organized. Take care of your materials and return them to their proper places. Sequence and Series Objectives: Define Sequence and Series. Generate pattern.  “The Gingerbread Man” MECHANICS: The students wil group into 5. The teacher wil read a story in front of the class. The students wil identify what are the arrangement of the setting on the story with the given pictures to every groups. The class wil only be given a 3 minutes to arrange the settings.  “GUESS THE MAGIC” MECHANICS: The students wil group into 5. The teacher wil present a pattern at the class. The students wil identify what are the next term.  “GUESS THE MAGIC” 1) 1, 5, 9, 13, 17, ____, _____, _____ 2) -12, -7, -2, 3, 8, ____, _____, ______ 3) 0, 15, 30, 45, _____, _____, ______ 4) 5, 10, 16, 23, 31, _____, ______, ______ 5) 2, 4, 6, _____, ______, ______ Sequence  SEQUENCE A sequence is a succession of numbers in a specific order. Each number in a sequence is called term.  SEQUENCE A sequence with a definite terms is a finite sequence. In a finite sequence, the first and last term are clearly identified. is a function whose domain is the set of natural numbers or a subset of consecutive positive integers.  Finite Sequence number of terms 1st term last term EXAMPLES: Identify the first term, last term, :and the number of terms: a.) 25, 17, 9, 1 b.) 0.5, 1.5, 4.5 c.) 5, 10  Infinite Sequence A sequence with no definite number of terms. example: 1.) -9, -2, 5, 12, 19,... 2.) 27, 9, 3, 1,…… EXAMPLES: Find the next three terms of each sequence: a.) 25, 17, 9,... b.) 0.5, 1.5, 4.5,... c.) 5, 10, 15, 25,... Use the functional notation F(n) =2n-3, where n is a natural number, to write an infinite sequence. Solution: If n = 1 If n = 2 If n = 3 F(1) = 2(1)-3 = 2-3 = -1  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 nth term of the sequence by using the term that precedes it.  EXPLICIT FORM A sequence can also be expressed in a form in which a preceding term is not necessary to find the succeeding terms. An explicit form can be used to find a term of the sequence by determining its position. RECURSIVE FORM Formula: 𝑎𝑛 = 𝑎𝑛−1 + 𝑑 EXPLICIT FORM Formula: 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑 Example: Find the next two terms in the given sequence; then write it in recursive form. 1. {7, 12, 17, 22, 27,…} 2. {3, 7, 15, 31, 63,…} Example: Determine the next two terms in the given sequence. Then write the explicit form. 1. {1, 4, 9, 16, 25,…} 2. {3, 5, 9, 17, 33,…} Series  SERIES It is the indicated sum of the terms of a sequence. A series can be denoted by Sn, where n refers to the number of terms. If a sequence is finite, its corresponding sries is finite series.  SERIES Formula: 𝑆𝑛 = 𝑎1 + 𝑎2 + 𝑎3 + … 𝑎𝑛 Where: S = series n = number of series Example: 3 8 15 24 The sum of the sequence 0, , , , 2 3 4 5  SUMMATION NOTATION The inconvenience of writing so many terms can be minimized using the summation notation. it makes use of the symbol (uppercase sigma), a greek letter equivalent to S, which is the first letter of the word sum. Example: Write each in expanded form. Then find its value. 6 5 a. b. ෍ 2𝑛 2 ෍ (𝑛 − 1) 𝑛=1 𝑛=3  ACTIVITY: Answer the following question on your book: Mental Math pp. 11 Example: