Warm Up Look for a pattern and predict the next number or expression in the list. 1. 1000, 500, 250, 125, _____ 2. 1, 2, 4, 7, 11, 16, _____ 3. 1, −3, 9, −27, _____ 4. 8, 3, −2, −7, _____ 5. 2, 2 2 , 4, 4 2, _____ 6. 7𝑎+4𝑏, 6𝑎+5𝑏, 5𝑎+6𝑏, 4𝑎+7𝑏, ______ 62.5 22 81 -12 8 3a+8b

Sequences and Series Chapter 13 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

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.  A geometric sequence is a sequence with the ratio between two consecutive terms constant.  This ratio is called the common ratio. 

Is each sequence arithmetic, geometric, or neither 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

Arithmetic Formula:       tn  =  t1  +  (n - 1)d tn is the nth term, t1 is the first term, n is the term number, and d is the common difference. Geometric Formula:        tn = t1 . r(n - 1) tn is the nth term, t1 is the first term, n is the term number, and r is the common ratio.

Find the first four terms and state whether the sequence is arithmetic, geometric, or neither. 1)  𝑡𝑛 = 3𝑛 + 2      2) 𝑡𝑛 = 𝑛2 + 1   3)  𝑡𝑛 = 3∙2𝑛 Find a formula for each sequence. 4) 2, 5, 8, 11, 14, . . . 5) 4, 8, 16, 32, . . . 6) 21, 201, 2001, 20001, . . .

Arithmetic Common difference = 3 1) 𝑡𝑛 = 3𝑛 + 2 Find the first four terms and state whether the sequence is arithmetic, geometric, or neither. 1)  𝑡𝑛 = 3𝑛 + 2      Arithmetic 1st Term: 𝑡1 = 3 1 + 2 =5 2nd Term: 𝑡2 = 3 2 + 2 =8 3rd Term: 𝑡3 = 3 3 + 2 =11 4th Term: 𝑡4 = 3 4 + 2 =14 Common difference = 3 First four terms: 5, 8, 11, 14

Find the first four terms and state whether the sequence is arithmetic, geometric, or neither. 1)  𝑡𝑛 = 3𝑛 + 2      2) 𝑡𝑛 = 𝑛2 + 1   3)  𝑡𝑛 = 3∙2𝑛 5, 8, 11, 14 Arithmetic d = 3 2, 5, 10, 17 Neither 6, 12, 24, 48 Geometric r = 2 Find a formula for each sequence. 4) 2, 5, 8, 11, 14, . . . 5) 4, 8, 16, 32, . . . 6) 21, 201, 2001, 20001, . . .

4) 2, 5, 8, 11, 14, . . . Arithmetic Find a formula for each sequence. 𝑡𝑛= 𝑡1+(𝑛 −1)𝑑 t1 = 2,  the first number in the sequence 𝑡𝑛=2+ 𝑛 −1 3 𝑡𝑛=2+3𝑛−3 d = 3, the common difference 𝑡𝑛=3𝑛−1

5) 4, 8, 16, 32, . . . Geometric Find a formula for each sequence. 𝑡𝑛= 𝑡 1 ∙ 𝑟 𝑛−1 t1 = 4,  the first number in the sequence 𝑡𝑛=4∙ 2 𝑛−1 𝑡𝑛=4∙ 2 𝑛 ∙ 2 −1 r = 2, the common ratio 𝑡𝑛=4∙ 2 𝑛 ∙ 1 2 𝑡𝑛= 2∙2 𝑛

It's not geometric or arithmetic. Find a formula for each sequence. It's not geometric or arithmetic. 6) 21, 201, 2001, 20001, . . . Think of the sequence as (20 +1), (200+1), (2000 + 1), (20000 + 1), . . . Then as this: [(2)(10) +1],[(2)(100) +1], [(2)(1000) +1], [(2)(10000) +1] Wait!  I see a pattern!   Powers of 10!  tn = 2.10n + 1 Does this work? Try it and see!

Find the indicated term of the arithmetic sequence with t1 = 5 and t7 = 29.    8) Find the number of multiples of 9 between 30 and 901.

Find the indicated term of the arithmetic sequence with t1 = 5 and t7 = 29.  Find t53 𝑡𝑛= 𝑡1+(𝑛 −1)𝑑 𝑡53=5+ 53 −1 4 29=5+(7 −1)𝑑 𝑡53=5+52∙4 29=5+6𝑑 24=6𝑑 𝑡53=213 𝑑=4

8) Find the number of multiples of 9 between 30 and 901. 36 What's the first multiple of 9 in the range? 900 What's the last multiple of 9 in the range? Use the arithmetic formula. 𝑡𝑛= 𝑡1+(𝑛 −1)𝑑 900= 36+9(𝑛−1) and solve for n 864=9𝑛−9 873=9𝑛 𝑛=97

Homework Page 476 #1-39

Challenge 1. Find t7 for an arithmetic sequence where t1 = 3x and d = -x. 2. Find t15 for an arithmetic sequence where t3 = -4 + 5i and t6 = -13 + 11i

2. Find  t15 for an arithmetic sequence where           t3 = -4 + 5i  and  t6 = -13 + 11i Challenge Answers 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 t3 to t15 is 13 terms. t15 = -4 + 5i + (13-1)(-3 +2i) = -4 + 5i -36 +24i      = -40 + 29i 1. Find t7 for an arithmetic sequence where                   t1 = 3x and d = -x. n = 7;  t1 = 3x, d = -x 𝑡 𝑛 = 𝑡 1 + 𝑛−1 𝑑 𝑡 7 =3𝑥+ 7−1 −𝑥 𝑡 7 =3𝑥+6 −𝑥 𝑡 7 =−3𝑥

Ch 13 sequences and series 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

notation Continuous function vs sequence or series Independent variable: x ⟹n Dependent variable: f(x) ⟹tn n is a subscript. Subscripts are counters. “t sub n”, means the value of term number n. Ex. t5=12, means the value of term 5, is 12. n is always a positive integer.