INTRODUCTION TO SEQUENCES

43210 In addition to level 3.0 and above and beyond what was taught in class, the student may: · Make connection with other concepts in math · Make connection with other content areas. The student will build a function (linear and exponential) that models a relationship between two quantities. The primary focus will be on arithmetic and geometric sequences. - Linear and exponential functions can be constructed based off a graph, a description of a relationship and an input/output table. - Write explicit rule for a sequence. - Write recursive rule for a sequence. The student will be able to: - Determine if a sequence is arithmetic or geometric. - Use explicit rules to find a specified term (n th ) in the sequence. With help from the teacher, the student has partial success with building a function that models a relationship between two quantities. Even with help, the student has no success understanding building functions to model relationship between two quantities. Focus 7 Learning Goal – (HS.F-BF.A.1, HS.F-BF.A.2, HS.F-LE.A.2, HS.F-IF.A.3) = Students will build a function (linear and exponential) that models a relationship between two quantities. The primary focus will be on arithmetic and geometric sequences.

SEQUENCE A sequence is an ordered list of elements. The elements of the list are called the terms of the sequence. (3, 5, 7, 9…) is a sequence. The “…” indicates that the pattern continues. We use subscript to identify each term of the sequence starting at 1:  a 1, a 2, a 3, a 4, …  a 1 = 3, a 2 = 5, a 3 = 7, a 4 = 9 a n is used to identify the n th term in the sequence.

TERM NOTATION Term Number Term Sequence Term Term a1a1 1 a2a2 2 a3a3 4 a4a4 8 a5a5 16 a6a6 32 Examine the two tables. Discuss with your partner what you notice about the two tables (similarities, differences). How do they relate to a “x, y” t-chart? How would you write a rule for this sequence?

WRITE A RULE FOR THE N TH TERM OF THE SEQUENCE This means that you need to write a rule that could calculate the 10 th term, the 99 th term or even the 150 th term of the sequence. Is this pattern linear or exponential? Since there isn’t a constant rate of change, it has to be exponential. 2 0, 2 1, 2 2, 2 3, 2 4, 2 5 … Sequence Term Term a1a1 1 a2a2 2 a3a3 4 a4a4 8 a5a5 16 a6a6 32

WRITE A RULE FOR THE N TH TERM OF THE SEQUENCE 2 0, 2 1, 2 2, 2 3, 2 4, 2 5 … What do you notice about the exponents and the sequence term numbers? The exponents are one less than the sequence term number. We can conclude that the rule would be: a n = 2 n-1 Sequence Term Term a1a1 1 a2a2 2 a3a3 4 a4a4 8 a5a5 16 a6a6 32 Pattern from:

EXPLICIT FORMULA An explicit formula allows you to find any element of a sequence without knowing the element before it. a n = 2 n-1 is an explicit formula. Use the rule to find the 10 th term of the sequence. a 10 = 2 (10-1) = 256 Pattern from:

WRITE A RULE (EXPLICIT FORMULA) FOR THE N TH TERM OF THE SEQUENCE 4, 7, 10, 13, 16, … Is this pattern linear or exponential? There is a constant rate of change (plus 3); therefore, it is linear. It helps to rewrite the sequence labeling each term with their term number. Since the sequence is plus 3, you need to multiply the term number by 3. What do you need to do next in order to get to the first number? Write the rule: a n = 3n + 1 Sequence Term Term a1a1 4 a2a2 7 a3a3 10 a4a4 13 a5a5 16

USE YOUR EXPLICIT FORMULA TO FIND THE 100 TH TERM OF THE SEQUENCE The rule is: a n = 3n + 1 Find a 100. a 100 = 3(100) + 1 a 100 = 301 Pattern from:

CREATE A SEQUENCE Consider the sequence generated by the formula a n = 6n – 4. Generate the first 5 terms of the sequence. a 1 = 6(1) – 4 a 2 = 6(2) – 4 a 3 = 6(3) – 4 a 4 = 6(4) – 4 a 5 = 6(5) – 4 2, 8, 14, 20, 26 Image from pple-apel-inti-makanan-urutan %2F&ei=GGGnVLPZEcffoASFhIK4Bg&bvm=bv ,d.cGU&psig=AFQjCNFGd2hazLmS3yHRW ytTWZQ&ust=