Exploring Sequences and Series in Mathematics
Delve into the world of sequences and series, understanding the concepts of arithmetic and geometric progressions. Learn to identify finite and infinite sequences, explore various patterns, and model them mathematically. From arithmetic to geometric, finite to infinite, this content offers a comprehensive overview for math enthusiasts.
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Sequences and Series Arithmetic Geometric ?1= first term ??= ?? 1 ? Where do I belong? ,?,?,?, ??=? 2(?1+ ??) ???????? ??????? ?1 ?1= first term ??= ?? 1+ ? ?1= first term ??=?1(1 ??) ???? ???? # ,?,?,?, ? = ?? ? =? + ? 1 ? ??= ?? 1+ ? ??= ?? 1 ? ????? ???? # ??= ?1 ?? 1 ? = ??= ?1+ (? 1) ? ?1= first term 1 ? ??= ?1 ?? 1 2 ??= ?1+ (? 1) ? ??=?1(1 ??) ??=? ???? ????? = ?????? ????????? = ??????????? 1 ? 2(?1+ ??) ?1 ? = ????? ????? = ???? ????? + 1 1 ? ,?,?,?, ? =? + ? ,?,?,?, ? = ?? Pg. 607 #1-32 2
Sequence and Series Describe Arithmetic Geometric Summation Modeling 10 10 10 10 10 20 20 20 20 20 30 30 30 30 30 40 40 40 40 40 50 50 50 50 50
Describe each using: Sequence/Series; Arithmetic/Geometric; Finite/Infinite 2, 8, 14, 20, 26 Answer Answer
Describe each using: Sequence/Series; Arithmetic/Geometric; Finite/Infinite 2, 8, 14, 20, 26 Finite Arithmetic Sequence
Describe each using: Sequence/Series; Arithmetic/Geometric; Finite/Infinite 2, 8, 32, 128, Answer Answer
Describe each using: Sequence/Series; Arithmetic/Geometric; Finite/Infinite 2, 8, 32, 128, Infinite Geometric Sequence
Describe each using: Sequence/Series; Arithmetic/Geometric; Finite/Infinite 1 + 5 + 9 + Answer Answer
Describe each using: Sequence/Series; Arithmetic/Geometric; Finite/Infinite 1 + 5 + 9 + Infinite Arithmetic Series
Describe each using: Sequence/Series; Arithmetic/Geometric; Finite/Infinite 16 + 8 + 4 + 2 + 1 Answer Answer
Describe each using: Sequence/Series; Arithmetic/Geometric; Finite/Infinite 16 + 8 + 4 + 2 + 1 Finite Geometric Series
Describe each using: Sequence/Series; Arithmetic/Geometric; Finite/Infinite 3(0.25)? ?=1 Answer Answer
Describe each using: Sequence/Series; Arithmetic/Geometric; Finite/Infinite 3(0.25)? ?=1 Infinite Geometric Series
Write in summation notation 31, 28, 25, 22,...,2 Answer Answer
Write in summation notation 31, 28, 25, 22,...,2 12 ??= ?1+ ? 1 ? 2 = 31 + ? 1 3 2 = 31 + 3? 3 2 = 34 + 3? 36 = 3? 12 = ? 31 + (? 1) 3 ?=1 Or 12 3? 34 ?=1
Write a recursive definition, explicit formula and find the 15 term for the sequence: 12, 21, 30, 39, . . . Answer Answer
Write a recursive definition, explicit formula and find the 15 term for the sequence: 12, 21, 30, 39, . . . Recursive: ?1= 12; ??= ?? 1+ 9 Explicit: ??= 12 + (? 1) 9 ?15= 12 + 15 1 9 = 138
Write a recursive definition, explicit formula and find the 18 term for the sequence: 45, 37, 29, 21, . . . Answer Answer
Write a recursive definition, explicit formula and find the 18 term for the sequence: 45, 37, 29, 21, . . . Recursive: ?1= 45; ??= ?? 1 8 Explicit: ??= 45 + (? 1) ( 8) ?18= 45 + 18 1 8 = 91
Find the missing term of the arithmetic sequence , 1, ,11,... Answer Answer
Find the missing term of the arithmetic sequence , 1, ,11,... = 1 + 11 = 5 2
Find the missing terms of the arithmetic sequence , 13, , , ,3,... Answer Answer
Find the missing terms of the arithmetic sequence 9, 1, , 13, , , ,3,... 5, = 13 + 3 = 5 2 = 13 + ( 5) = 9 2 = 5 + 4 = 1
Write in summation notation 5,10,20,40, ,640 Answer Answer
Write in summation notation 5,10,20,40, ,640 ??= ?1 ?? 1 640 = 5 2? 1 128 = 2? 1 ? 1 = log2128 ? 1 = 7 ? = 8 8 5 2? 1 ?=1
Write a recursive definition, explicit formula and find the 10 term for the sequence: 3,6, 12,24,... Answer Answer
Write a recursive definition, explicit formula and find the 10 term for the sequence: 3,6, 12,24,... Recursive: ?1= 3; ??= ?? 1 2 Explicit: ??= ( 3) ( 2)? 1 ?10= ( 3) ( 2)10 1= 1536
Write a recursive definition, explicit formula and find the 9 term for the sequence: 2, 10, 50, 250,... Answer Answer
Write a recursive definition, explicit formula and find the 9 term for the sequence: 2, 10, 50, 250,... Recursive: ?1= 2; ??= ?? 1 5 Explicit: ??= ( 2) (5)? 1 ?9= 2 59 1= 781250
Find the missing term of the geometric sequence ,3, ,12,... Answer Answer
Find the missing term of the geometric sequence ,3, ,12,... = 12 3 = 6
Find the missing terms of the geometric sequence , 20, , , , 1.25,... Answer Answer
Find the missing terms of the geometric sequence 10, , 1.25,... 2.5, , 20, , , 5, = 20 1.25 = 5 = 20 5 = 10 = 5 1 2= 2.5
Evaluate the sum of each series 5 + 11 + 17 + + 35 Answer Answer
Evaluate the sum of each series 5 + 11 + 17 + + 35 ??= ?1+ (? 1) ? 35 = 5 + ? 1 6 35 = 5 + 6? 6 35 = 6? 1 36 = 6? 6 = ? ??=? 2(?1+ ??) ?6=6 2(5 + 35) = 120
Evaluate the sum of each series 4 + 16 + 64 + + 4096 Answer Answer
Evaluate the sum of each series 4 + 16 + 64 + + 4096 ??= ?1 ?? 1 4096 = 4 4? 1 1024 = 4? 1 ? 1 = log41024 ? 1 = 5 ? = 6 ??=?1(1 ??) 1 ? ?6=4(1 46) 1 4 = 5460
Evaluate the sum of each series 15 7? 5 ?=4 Answer Answer
Evaluate the sum of each series 15 7? 5 ?=4 ??=? 2(?1+ ??) ?4= 7 4 5 = 23 ?15= 7 15 5 = 100 ?12=12 2(23 + 100) = 738
Evaluate the sum of each series 7 4(5)? 1 ?=1 Answer Answer
Evaluate the sum of each series 7 4(5)? 1 ?=1 ??=?1(1 ??) ?1= 4(5)1 1= 4 1 ? ?6=4(1 57) 1 5 = 78124
Evaluate the sum of each series 3 3?2 ?=1 Answer Answer
Evaluate the sum of each series 3 3?2 ?=1 ?1= 3(1)2= 3 ?2= 3(2)2= 12 ?3= 3(3)2= 27 + 42
The number of toy rockets made by an assembly line for 8 hours forms an arithmetic sequence. If the line produced 40 rockets in hour one and 43 rockets in hour two, how many rockets will be produced in hour seven? How many rockets will be produced in one 8 hour day? Answer Answer
The number of toy rockets made by an assembly line for 8 hours forms an arithmetic sequence. If the line produced 40 rockets in hour one and 43 rockets in hour two, how many rockets will be produced in hour seven? How many rockets will be produced in one 8 hour day? ??= ?1+ (? 1) ? ??= 40 + ? 1 3 ??=? ?8=8 2(?1+ ??) 2(40 + 61) = 404 ?7= 40 + 7 1 3 = 58
You invested money in a fund and each month you receive a payment for your investment. Over the first four months, you received $50, $57, $64, $71. If this pattern continues, how much will you receive in the 12th month and who much will you receive for the entire year? Write an explicit equation to model the problem. Answer Answer
You invested money in a fund and each month you receive a payment for your investment. Over the first four months, you received $50, $57, $64, $71. If this pattern continues, how much will you receive in the 12th month and who much will you receive for the entire year? Write an explicit equation to model the problem. ??=? 2(?1+ ??) ??= ?1+ (? 1) ? ??= 50 + ? 1 7 ?8=12 2(50 + 127) ?12= 50 + 12 1 7 = 127 = 1062
You are trying to save $1500. You begin with $5 and save $3 more than the previous week for 30 weeks. Will you meet your goal? Write an explicit formula to model this problem What is the amount you will save in week 30? What is the total amount you will save over 30 weeks? Answer Answer
You are trying to save $1500. You begin with $5 and save $3 more than the previous week for 30 weeks. Will you meet your goal? Write an explicit formula to model this problem What is the amount you will save in week 30? What is the total amount you will save over 30 weeks? No ??=? 2(?1+ ??) ??= ?1+ (? 1) ? ??= 5 + ? 1 3 ?30=30 2(5 + 92) = 1455 ?30= 5 + 30 1 3 = 92
You saved $500 this year. Each year you plan to save 5% more than the previous year. Write an explicit formula to model this situation. How much will you save in the 8 year? How much will you have saved totally over years, assume you do not spend anything? Answer Answer
You saved $500 this year. Each year you plan to save 5% more than the previous year. Write an explicit formula to model this situation. How much will you save in the 8 year? How much will you have saved totally over years, assume you do not spend anything? ??=?1(1 ??) ??= ?1 ?? 1 ??= 500 (1.05)? 1 1 ? ?8=500(1 1.058) 1 1.05 ?8= 500 (1.05)8 1 = 703.55 = 4774.55
