, 500, 250, 125, , 2, 4, 7, 11, 16, , 3, 9, 27, , 3, 2, 7, , 2 2, 4, 4 2, 8

Transcription

1 Warm Up Look for a pattern and predict the next number or expression in the list , 500, 250, 125, , 2, 4, 7, 11, 16, , 3, 9, 27, , 3, 2, 7, , 2 2, 4, 4 2, a + 4b, 6a + 5b, 5a + 6b, 4a + 7b, 3a+8b

2 13.1 Arithmetic and Geometric Series Objective:To identify an arithmetic or geometric sequence and find a formula for its nth term. Chapter 13 Sequences and Series

3 A sequence is a set of numbers, called terms, arranged in some particular order. An arithmetic sequence is a sequence with the difference between two consecutive terms constant. The difference is called the common difference, d. A geometric sequence is a sequence with the ratio between two consecutive terms constant. This ratio is called the common ratio, r.

4 Is each sequence arithmetic, geometric, or neither? What is the common difference or common ratio? 1) 3, 8, 13, 18, 23,... 2) 1, 2, 4, 8, 16,... 3) 24, 12, 6, 3, 3/2, 3/4,... 4) 55, 51, 47, 43, 39, 35,... 5) 2, 5, 10, 17,... 6) 1, 4, 9, 16, 25, 36,... 7) 3, 3, 3, 3, 3, 1) Arithmetic, d = 5 2) Geometric, r = 2 3) Geometric, r = 1/2 4) Arithmetic, d = -4 5) Neither 6) Neither 7) Either, d = 0, r = 1

5 Arithmetic Formula: t n = t 1 + (n - 1)d (discrete linear) Or: t n = dn + t 1 - d t n is the nth term, t 1 is the first term, n is the term number, and d is the common difference. Geometric Formula: t = t. r (n - 1) n 1 (discrete exponential) Or: t n = t 1 r. r n t n is the nth term, t 1 is the first term, n is the term number, and r is the common ratio.

6 Is each sequence arithmetic, geometric, or neither? What is the common difference or ratio? What is the formula? 1) 3, 8, 13, 18, 23,... 2) 1, 2, 4, 8, 16,... 3) 24, 12, 6, 3, 3/2, 3/4,... 4) 55, 51, 47, 43, 39, 35,... 5) 3, 3, 3, 3, 3, 1) A, d = 5 t n = 5n 2 2) G, r = 2 t = 1. 2 n n 2 3) G, r = 1/2 t n = 48( 1 2 )n 4) A, d = -4 t n = 4n ) A, d=0 t n = 3 G, r=1 t n = 3

7 Find the first four terms and state whether the sequence is arithmetic, geometric, or neither. 1) t n = 3n + 2 2) t n = n ) t n = 3 2 n Find a formula for each sequence. 4) 2, 5, 8, 11, 14,... 5) 4, 8, 16, 32,... 6) 21, 201, 2001, 20001,...

8 Find the first four terms and state whether the sequence is arithmetic, geometric, or neither. 1) t n = 3n st Term: t 1 = = 5 2 nd Term: t 2 = = 8 3 rd Term: t 3 = = 11 4 th Term: t 4 = = 14 Arithmetic Common difference = 3 First four terms: 5, 8, 11, 14

9 Find the first four terms and state whether the sequence is arithmetic, geometric, or neither. 1) t n = 3n + 2 2) t n = n ) t n = 3 2 n Find a formula for each sequence. 4) 2, 5, 8, 11, 14,... 5) 4, 8, 16, 32,... 6) 21, 201, 2001, 20001,... 5, 8, 11, 14 Arithmetic d = 3 2, 5, 10, 17 Neither 6, 12, 24, 48 Geometric r = 2

10 Find a formula for each sequence. 4) 2, 5, 8, 11, 14,... t 1 = 2, the first number in the sequence d = 3, the common difference Arithmetic t n = dn + t 1 d t = 3n n t = 3n 1 n

11 Find a formula for each sequence. 5) 4, 8, 16, 32,... t 1 = 4, the first number in the sequence r = 2, the common ratio Geometric t n = t 1 r rn t n = 4 2 2n t n = 2 2 n

12 Find a formula for each sequence. 6) 21, 201, 2001, 20001,... It's not geometric or arithmetic. Think of the sequence as (20 +1), (200+1), ( ), ( ),... Then as this: [(2)(10) +1],[(2)(100) +1], [(2)(1000) +1], [(2)(10000) +1] Wait! I see a pattern! Powers of 10! t n = n + 1 Does this work? Try it and see!

13 7) Find the indicated term of the arithmetic sequence with t 1 = 5 and t 7 = 29. Find t 53 8) Find the number of multiples of 9 between 30 and 901.

14 7) Find the indicated term of the arithmetic sequence with t 1 = 5 and t 7 = 29. Find t 53 t = t + (n 1)d n 1 29 = 5 + (7 1)d 29 = 5 + 6d 24 = 6d t 53 = t 53 = t 53 = 213 d = 4

15 8) Find the number of multiples of 9 between 30 and 901. What's the first multiple of 9 in the range? What's the last multiple of 9 in the range? Use the arithmetic formula. t = t + (n 1)d n = (n 1) and solve for n 864 = 9n = 9n n = 97

16 Challenge 1. Find t 7 for an arithmetic sequence where t 1 = 3x and d = -x. 2. Find t 15 for an arithmetic sequence where t 3 = i and t 6 = i

17 Challenge Answers 1. Find t 7 for an arithmetic sequence where t 1 = 3x and d = -x. n = 7; t 1 = 3x, d = -x t n = t 1 + n 1 d t 7 = 3x x t 7 = 3x + 6 x t 7 = 3x 2. Find t 15 for an arithmetic sequence where t 3 = i and t 6 = i Get a visual image of this problem Using the third term as the "first" term, find the common difference from these known terms. Now, from t 3 to t 15 is 13 terms. t 15 = i + (13-1)(-3 +2i) = i i = i

18 Homework Page 476 #1-39 odds

19 Sequences and Series Chapter 13 Learn notation to define sequences, series, sums of series, and specific terms of either. Identify, find formulas, and find specific terms of sequences Find sums, and formulas for sums, for finite series Determine if an infinite series has a limit If so, find the sum of the series