Recursive Sequences. EQ: How do I write a sequence to relate each term to the previous one?

Transcription

1 Recursive Sequences EQ: How do I write a sequence to relate each term to the previous one? Dec 14 8:20 AM Arithmetic Sequence - A sequence created by adding and subtracting by the same number known as the common difference "d". Geometric Sequence - A sequence created by multiplying and dividing by the same number known as the common ratio "r". Dec 14 8:35 AM 1

2 Recursive Rule - a sequence that tells you a term and relates each additional term to the previous ones. Explicit/Closed Rule - a sequence that is defined by the number of the term in the sequence that you're on. Dec 14 8:35 AM Two ways to define 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 Closed: can directly find n th term NOT required to list 1 st term Aug 26 4:04 PM 2

3 Dec 16 12:31 PM Writing a Recursive Sequence 9, 6, 3, 0, 3 a 1 = a n 1 Dec 16 11:06 AM 3

4 Finding a Sequence from a Rule a 1 = 5 3a n 1 Dec 16 11:06 AM Recursive definition: Ex 1: 5, 10, 15, 20, a 1 = Ex 2: 2, 4, 8, 16, a 1 = Aug 26 4:10 PM 4

5 Recursive definition: Ex 3: 2, 8, 14, 20, a 1 = Ex 4: 81, 27, 9, 3, 1... a 1 = Aug 26 4:10 PM Recursive definition: Ex 5: 5, 2, 1, 4, 7... a 1 = Ex 6: 16, 48, 144, 432, a 1 = Aug 26 4:10 PM 5

6 Ex 7: Given the recursive definition, find the first five terms of the sequence. a 1 = 3 a n Ex 8: Given the recursive definition, find the first five terms of the sequence. a 1 = 8 a n 1 3 6

7 Ex 9: Given the recursive definition, find the first five terms of the sequence. a 1 = 3 2a n 1 Ex 10: Given the recursive definition, find the first five terms of the sequence. a 1 = 81 2/3a n 1 7

8 Explicit/Closed Form of Arithmetic Sequences EQ: How do I write a formula to go directly to a term in a sequence? Dec 16 3:46 PM Types of sequences: Arithmetic - the change is adding or subtracting the same number. We call this the common difference "d". Geometric - the change is multiplying by the same number. We call this the common ratio "r". Sep 23 11:16 PM 8

9 Two ways to define 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 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 and a 1 r n 1 for geometric Aug 26 4:04 PM Writing the closed/explicit form of an arithmetic sequence: Step 1: Find the difference "d" between each term. Step 2: Subtract this "d" from the first term to find a 0. Step 3: Plug into form dn + a 0 (like y = mx + b) Dec 16 3:47 PM 9

10 Domain and Range of a Sequence: Domain: The whole numbers of the terms in your sequence. Range: The terms of the sequence. Example: 3, 5, 7, 9, 11 Domain: {1, 2, 3, 4, 5} (5 terms in the sequence) Range: {3, 5, 7, 9, 11} (same as the sequence itself) Dec 16 3:49 PM Looks like a linear function with the difference related to the slope. y = mx + b dn + a 0 "a 0 " is the term 1 before the sequence and "d" is the common difference. 9, 7, 5, 3, 1 Domain: Range: Dec 16 11:14 AM 10

11 Closed/Explicit definition: Ex 1: 5, 10, 15, 20, Ex 2: 2, 4, 10, 16, Aug 26 4:10 PM Closed/Explicit definition: Ex 3: 2, 8, 14, 20, Ex 4: 30, 25, 20, 15, Aug 26 4:10 PM 11

12 Closed/Explicit definition: Ex 5: 5, 2, 1, 4, 7... Ex 6: 5, 5, 15, 25, Aug 26 4:10 PM Ex 7: Given the explicit definition, find the first five terms of the sequence. Then, find the 20th term. 4n 1 12

13 Ex 8: Given the explicit definition, find the first five terms of the sequence. Then find the 20th term. 3n + 8 Ex 9: Given the explicit definition, find the first five terms of the sequence. Then find the 20th term. 2n 13

14 Ex 10: Given the explicit definition, find the first five terms of the sequence. Then find the 20th term. 3n 2 n Explicit/Closed Form of Geometric Sequences EQ: How do I write a formula to go directly to a term in a sequence? Dec 16 3:46 PM 14

15 Types of sequences: Arithmetic - the change is adding or subtracting the same number. We call this the common difference "d". Geometric - the change is multiplying by the same number. We call this the common ratio "r". Sep 23 11:16 PM Two ways to define 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 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 and a 1 r n 1 for geometric Aug 26 4:04 PM 15

16 Writing the closed/explicit form of an geometric sequence: Step 1: Find the ratio "r" between each term. Step 2: Find the first term a 1. Step 3: Plug into form a 1 (r) n-1 (like y = a(b) x ) Dec 16 3:47 PM Domain and Range of a Sequence: Domain: The whole numbers of the terms in your sequence. Range: The terms of the sequence. Example: 2, 4, 8, 16, 32 Domain: {1, 2, 3, 4, 5} (5 terms in the sequence) Range: {2, 4, 8, 16, 32} (same as the sequence itself) Dec 16 3:49 PM 16

17 Looks like an exponential function with the ratio related to the base. y = A(B) x a 1 (r) n 1 "a 1 " is the first term and "r" is the common ratio. 1, 3, 9, 27, 81 Domain: Range: Dec 16 11:14 AM Closed/Explicit definition: Ex 1: 2, 10, 50, 250, Ex 2: 2, 4, 8, 16, Aug 26 4:10 PM 17

18 Closed/Explicit definition: Ex 3: 2, 2, 2, 2, 2... Ex 4: 3, 6, 12, 24, Aug 26 4:10 PM Closed/Explicit definition: Ex 5: 625, 375, 225, 135, Ex 6: 90, 30, 10, 3.33, Aug 26 4:10 PM 18

19 Ex 7: Given the explicit definition, find the first five terms of the sequence. Then, find the 20th term. 4(2) n 1 Ex 8: Given the explicit definition, find the first five terms of the sequence. Then find the 10th term. 3(8) n 1 19