Arithmetic Sequences

Transcription

1 Vocabulary: Arithmetic Sequence a pattern of numbers where the change is adding or subtracting the same number. We call this the common difference "d". Closed/Explicit Formula a formula for a sequence that is defined by the number of the term and goes directly to that term. Recursive Formula a sequence of numbers created by defining a term in the sequence and the pattern created by the sequence using previous terms. Subscript a number on a variable that identifies which variable it is. Example: x 2 means the second x. Sep 23 11:16 PM Arithmetic Sequences F.BF.1a Determine an explicit expression and the recursive process (steps for calculation) from context. F.BF.2 Write arithmetic and geometric sequences recursively and explicitly, use them to model situations, and translate between the two forms. Connect arithmetic sequences to linear functions and geometric sequences to exponential functions. F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Dec 16 3:46 PM 1

2 What am I learning today? How to recognize and write the formula for an arithmetic sequence How will I show that I learned it? Write a recursive and an explicit formula for an arithmetic sequence Aug 18 1:21 PM Two ways to define sequences: Closed/Explicit: can directly find n th term NOT required to list 1 st term uses formula a 1 + d(n 1) for arithmetic sequences Recursive: relates each term in the seq. to a previous term must ALWAYS state 1 st term requires formula that relates the n th term to the (n 1) th term Aug 26 4:04 PM 2

3 Arithmetic Sequences Sequences that are created by ADDING OR SUBTRACTING the same value. We call this value the COMMON DIFFERENCE. When graphed, it looks like a LINEAR FUNCTION with the difference related to the slope. Example: 9, 7, 5, 3, 1,... Oct 23 11:12 AM Parts of a Sequence 9, 7, 5, 3, 1,... Oct 23 11:12 AM 3

4 Domain and Range of a Sequence: Domain: The whole numbers of the terms in your sequence. Range: The terms of the sequence. Example: 9, 7, 5, 3, 1 Domain: {,,,, } (5 terms in the sequence) Range: {,,,, } (same as the sequence itself) Dec 16 3:49 PM Steps for finding the closed/explicit form: Option 1 Option 2 Use Formula Treat like a Linear Function Formula: a 1 + d(n 1) 1) Plug in a 1 and d from your sequence. 2 ) Distribute your d and combine like terms. 1) Find the difference "d" between each term. 2) Subtract this "d" from the first term to find a 0. Example: 9, 7, 5, 3, 1,... 3) Plug into form dn + a 0 (like y = mx + b) Example: 9, 7, 5, 3, 1,... Dec 16 3:47 PM 4

5 Closed/Explicit definition: Ex 1: 5, 10, 15, 20, Ex 2: 2, 4, 10, 16, D: R: D: R: Aug 26 4:10 PM Closed/Explicit definition: Ex 3: 2, 8, 14, 20, Ex 4: 30, 25, 20, 15, D: R: D: R: Aug 26 4:10 PM 5

6 Ex 5: Given the explicit definition, find the first five terms of the sequence. Then, find the 20th term. 4n 1 Aug 26 4:14 PM Ex 6: Given the explicit definition, find the first five terms of the sequence. Then find the 20th term. 3n + 8 Aug 26 4:14 PM 6

7 Ex. 7 a 3 = 6 d = 4 What is the 20th term of the sequence? Oct 24 10:50 AM Ex. 8 a 4 = 12 a 7 = 27 What is the 200th term of the sequence? Oct 23 11:27 AM 7

8 Parts of a Recursive Definition a 1 = 8 a n Oct 27 10:44 AM Evaluating a Recursive Definition a 1 = 8 a n Oct 27 10:44 AM 8

9 Ex 1: Given the recursive definition, find the first four terms of the sequence. a 1 = 7 a n 1 3 Aug 26 4:14 PM Ex 2: Given the recursive definition, find the first four terms of the sequence. a 1 = 5 2a n 1 Aug 26 4:14 PM 9

10 Ex 3: Given the recursive definition, find the first five terms of the sequence. a 1 = 3 a n Aug 26 4:14 PM Understanding a Recursive Definition n a n a n 1 + 2a n 3 Find the missing terms a 4 a 7 a 9. Oct 27 10:44 AM 10

11 Writing a Recursive Definition 1) Identify the first term and label it a 1. This is PART OF THE DEFINITION. 2) Identify the pattern. Write an equation that represents the pattern using a n as the current term, a n 1 as the last term, a n 2 as 2 terms ago, etc. Example A: 3, 6, 12, 24, 48 Example B: 5, 1, 3, 7, 11 Oct 27 10:44 AM Recursive definition: Ex 1: 5, 10, 15, 20, a 1 = Aug 26 4:10 PM 11

12 Recursive definition: Ex 2: 2, 8, 14, 20, a 1 = Aug 26 4:10 PM Recursive definition: Ex 3: 5, 2, 1, 4, 7... a 1 = Aug 26 4:10 PM 12

13 Extension: Fibonacci Sequence a 1 = 1, a 2 = 1 a n-1 + a n-2 Oct 27 11:00 AM 13