= n which will be written with general term a n Represent this sequence by listing its function values in order:

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1 Sectio 9.: Sequeces a Series (Sums) Remier:, 2, 3,, k 2, k, k, k +, k + 2, The ellipsis ots iicate the sequece is oeig, has ifiitely may terms. I. SEQUENCES Defiitio (page 706): A sequece is a fuctio whose omai is the set of positive itegers (coutig umbers, 2, 3,...) Example: Compare the fuctio f ( x) = x to the x + sequece f ( ) = which will be writte with geeral + term a =. + Represet this sequece by listig its fuctio values i orer: f (), f (2), f (3), f (4),...,,,, Istea of fuctio otatio, represet a sequece by subscripte letters: a, a a, a,..., 2, 3, 4,... 2, The values i the orere list are the terms of the sequece. Do t cofuse term umbers (omai values) with term values.

2 Sectio 9.: Sequeces a Series (Sums) The subscript represets the term umber a a represets the term value. This sequece is give by the formula{ a} = or more simplya + =. + Vocabulary: preceig, succeeig, Give the geeral term, fi the sequece examples: th term First five terms 00 th term a= 2 a 2 = a = 2 a 2 = ( ) a = + ( ) + a = + Give the sequece, fi the geeral term examples: First few terms, ext term 00 th term th term 3, 6, 9, 2, 5,,,,, ,,,, , 5, 6, 7,

3 Sectio 9.: Sequeces a Series (Sums) Defiitio (page 708) Factorial: If is a whole umber,! is the prouct of the whole umbers from up to :! = 2 3 ( 2)( ) 0! = a! = Examples with Factorials: 7! = List the first five terms of the sequece: a x =! 7! 9! =! = ( + 3)! Defiitio: A sequece whose th term epes o oe or more of the terms preceig it is recursive. Examples of Recursive Sequeces: a = 2, a = 2 a + 5 The Fiboacci Sequece,, 2, 3, 5, 8,... is efie recursively as F =, F 2 = af = F + F 2 3

4 Sectio 9.: Sequeces a Series (Sums) II. SERIES (SUMS) I calculus, terms of a sequece are ae to compute itegrals. Partial sums for the sequece a, a, a, a,..., a..., S k = a (st partial sum) S = a + a (2 partial sum) 2 2 S = a + a + a (3r partial sum) S = a + a + a + a (4th partial sum) S = a + a + a + a a + a (th partial sum) S = a a a (ifiite sum) We ca also write the partial sum usig sigma otatio S = a + a + a + a a + a = a k= The Greek capital letter sigmaσ iicates aitio. The iteger k is calle the iex of summatio. The lower limit of summatio is ; the upper limit of summatio is. Example: For, 3, 9, 27,... fi S a 4 S. 6 Example: Write out the sums 6 k k 7 = + a 4 ( ). = 0! Page 74 66, 68, 70, 74 k 4

5 Sectio 9.: Arithmetic Sequeces a Series I. ARITHMETIC SEQUENCES Examples: These sequeces are arithmetic: 5,,3,7,... a 5,2,, 4, 7,... Defiitio: A sequece is arithmetic whe the ifferece betwee successive terms is a costat : a a = a a =... = a a = + A arithmetic sequece is efie recursively as: a = a a a = a + (page 77) + The commo ifferece is fou by takig ay term a subtractig its preecessor= a a. + Here are the first few terms of a arithmetic sequece: a = a a 2 = a + = a + a = a + = a + + = a a = a + = a = a a = a + = a = a The geeral patter is a= a + = a + ( ) th term of a Arithmetic Sequece: Page 723 8, 0, 8, 20, 26, 32, 38 a= a + ( ) 5

6 Sectio 9.2: Arithmetic Sequeces & Series II. ARITHMETIC SERIES (SUMS) To fi the sum of terms of a arithmetic sequece (page 70), use oe of two formulas: S= ( a + a ) 2 or S= 2 a + ( ) 2 Where o these formulas come from? Page , 50 6

7 Sectio 9.3: Geometric Sequeces a Series GEOMETRIC SEQUENCES Examples: These sequeces are geometric: 2, 2, 6,8, a 6, 2, 2, 2, 2, Defiitio: A sequece is geometric whe the ratio betwee successive terms is a costat r. A arithmetic sequece is efie recursively as: a = a a a ra + = The commo ratio r is fou by takig ay term a iviig by its preecessor r= a a. + / Here are the first few terms of a geometric sequece: a = a = a r0 a = ra = a r 2 a = ra = r( a r) = a r2 3 2 = = ( 2) = a ra r a r a r The geeral patter is a = ra = a r This allows us to efie a geometric sequece by its th term: a = a r (page 727) Page 735 4, 6, 4, 6, 24, 30 7

8 Sectio 9.3 Geometric Sequeces & Series II. GEOMETRIC SERIES To fi the sum of terms of a geometric sequece (page 597): S = a r = a + a r+ a r + a r +... a r = a k= Page , 36 k 2 3 ( r) ( r) If the terms of the sequece are gettig small (approachig zero) quickly eough, a ifiite series will sometimes approach a umber. A ifiite geometric series a + a r+ a r2+ a r3+... a ri = will i= approach a limitig umber as log as r is a umber with absolute value smaller tha. The sum of a ifiite geometric series is S a = = a r if = 0 r Page , 50 r<. (Page 73) 8

Unit 6: Sequences and Series

Unit 6: Sequences and Series AMHS Hoors Algebra 2 - Uit 6 Uit 6: Sequeces ad Series 26 Sequeces Defiitio: A sequece is a ordered list of umbers ad is formally defied as a fuctio whose domai is the set of positive itegers. It is commo