Unit 8 Arithmetic Sequences and Series and Geometric Sequences and Series

University:
University of California, Berkeley
Course:
MATH W54 | Linear Algebra and Differential Equations
Academic year:

2021
Views:

139

Pages:

3
Author:

maselarmwpa

Unit 8 Arithmetic Sequences/Series and Geometric Sequences/Series ● A sequence is a function whose domain is the set of positive functions. 1 ● The terms for the sequence 𝑓(𝑛) = 𝑛 ○ n = input, position ○ fn = term ● Factorials ○ 𝑛! = 𝑛(𝑛 − 1)(𝑛 − 2). . .3 ⋅ 2 ⋅ 1where 0! = 1and 1!=1 ● Recursive Formulas ○ A sequence that uses previous terms ● Properties of Sequences ○ ∑𝑛𝑘=1 𝑐 = 𝑐 + 𝑐 + 𝑐+. . . +𝑐 = 𝑐𝑛 ○ ∑𝑛𝑘=1 (𝑐𝑎𝑘 ) = 𝑐𝑎1 + 𝑐𝑎2 + 𝑐𝑎3 +. . . +𝑐𝑎𝑛 = 𝑐(𝑎1 + 𝑎2 +. . . +𝑎𝑛 ) = 𝑐 ∑𝑛𝑘=1 𝑎𝑘 ○ ∑𝑛𝑘=1 (𝑎𝑘 + 𝑏𝑘 ) = ∑𝑛𝑘=1 (𝑎𝑘 ) + ∑𝑛𝑘=1 (𝑏𝑘 ) ○ ∑𝑛𝑘=1 (𝑎𝑘 − 𝑏𝑘 ) = ∑𝑛𝑘=1 (𝑎𝑘 ) − ∑𝑛𝑘=1 (𝑏𝑘 ○ ∑𝑛𝑘=1 (𝑎𝑘 ) = ∑𝑘=1 ○ ∑𝑛𝑘=1 𝑘 = 1 + 2 + 3+. . . +𝑛 = ○ ∑𝑛𝑘=1 𝑘 2 = 12 + 22 + 32 +. . . +𝑛2 = ○ ∑𝑛𝑘=1 𝑘 3 = 13 + 23 + 33 +. . . +𝑛3 = ( 𝑗 (𝑎𝑘 ) + ∑𝑛𝑘=𝑗+1 (𝑏𝑘 ); 0 < 𝑗 < 𝑛 𝑛(𝑛+1) 2 𝑛(𝑛+1)(2𝑛+1) 6 𝑛(𝑛+1) 2 ) 2 ● Example: ○ ∑9𝑘=1 (3𝑘) = 3 ∑9𝑘=1 𝑘 =3⋅( 9(9+1) 2 ) = 135 ● Arithmetic Sequences: differences between successive terms is always the same number ○ a1 = a ○ an = an-1 + d ■ d: common difference(slope) ■ a: first term ● Example: 4,7,10,13,... ○ 7 - 3 =3 ○ 10 - 7 = 3 ○ 13 - 10 = 3 ○ Answer: ■ d=3&a=4 ● Arithmetic Sequence Theorem: an = a + (n - 1)d ● Example: find the 13th term of the sequence: 2,6,10,14,18… ○ d=4, a=2 ○ a13 = 2 + (13 - 1)4 = 50 ● Arithmetic Series Theorem: Let {an} be an arithmetic sequence with first term a and a common difference d. The sum Sn of the first n terms of {an} is 𝑛 𝑛 ○ 𝑆𝑛 = 2 [2𝑎 + (𝑛 − 1)𝑑] = 2 (𝑎 + 𝑎𝑛 ) ● Find the sum Sn of the first n terms of the following sequence (in terms of ‘n’) ○ {3n + 5} 𝑛 𝑛 ○ 𝑆𝑛 = 2 [8 + 3𝑛 + 5] = 2 (3𝑛 + 13) = ○ Answer: 3𝑛2 +13𝑛 2 3𝑛2 +13𝑛 2 ● Geometric Sequence: ratio of successive terms is the same nonzero number ○ a1 = a ■ a = 1st term ○ an = ran-1 ■ r = common ratio ● Geometric Sequence Theorem: For a geometric sequence {an} whose first term is a and whose common ratio is r, the n-th term is determined by ○ an = arn-1 ● Adding the first n terms of a geometric sequence ○ Sn = 𝑎 1−𝑟 𝑛 1−𝑟 ● Infinite Geometric Series Theorem ○ The sum of an infinite geometric series is: ■ ∑∞ 𝑘=1 𝑎 𝑎𝑟 𝑘−1 = 1−𝑟