Algebra 2 AII.2 Geometric Sequences and Series Notes. Name: Date: Block:

Transcription

1 Algeba 2 AII.2 Geometic Sequeces ad Seies Notes Ms. Giese Name: Date: Block: Geometic Sequeces Geometic sequeces cotai a patte whee a fixed amout is multiplied fom oe tem to the ext (commo atio ) afte the fist tem Geometic sequece examples: o 2, 4, 6, 32, o Domai: o Rage: o Gaph show at ight o commo atio = o The gaph of a geometic sequece is o Fid the commo atio () fo the followig geometic sequeces: a) 5, 0, 20, 40, = b) -, 22, -44, 88,. = c) 4,,,,, = Idetifyig Geometic Sequeces o Idetify whethe the followig sequeces ae aithmetic, geometic, o eithe. If it is aithmetic, fid d ad if it is geometic, fid. a) 4, 0, 8, 28, 40, b) 625, 25, 25, 5,, c) 8, 27, 9, 3,, d), 2, 6, 24, 20, e) -4, 8, -6, 32, -64, f) 8,, -6, -3, -20,

2 Algeba 2 AII.2 Geometic Sequeces ad Seies Notes Ms. Giese Page 2 Fidig Tems i a Geometic Sequece o Fid the 7 th tem i the sequece: 2, 6, 8, 54, = a7 = o Is thee a patte? a = 2 a2 = a a3 = a2 = a = a4 = a3 = a2 = a = a = To fid the th tem i a geometic sequece: a = a whee a is the fist tem of the sequece, is the commo atio, is the umbe of the tem to fid o You ty a) Fid the commo atio : 3 3 6, -3,,, 2 4 b) Fid the commo atio fo the sequece give by the fomula a=5(3) - c) Fid the 7 th tem of the sequece: 2, 6, 8, 54, d) Fid a8 fo the sequece 0.5, 3.5, 24.5, 7.5,... e) Wite a ule fo the th tem of the sequece, the fid a7 4, 20, 00, 500,... f) Oe tem of a geometic seies is a4=2. The commo atio =2. Wite a ule fo the th tem. g) Two tems i a geometic sequece ae a3 = -48 ad a6 = Fid a ule fo the th tem.

3 Algeba 2 AII.2 Geometic Sequeces ad Seies Notes Ms. Giese Page 3 Geometic Seies A geometic seies is the sum of the tems i a geometic sequece: S = a Sums of a Fiite Geometic Seies i i o The sum of the fist tems of a geometic seies is give by: S = a whee a is the fist tem i the sequece, is the commo atio, ad is the umbe of tems to sum. o Why? Expad S = Multiply both sides by : Subtact: Solve fo S: o Examples: a) Fid the sum: 6 i 4(3) i b) Fid the sum of the fist 8 tems of the sequece: -5, 5, -45, 35, c) Fid the sum: 5 k 3 k o You Ty a) Fid the sum: 8 k 6( 2) k b) Fid the sum of the fist 8 tems of the sequece: 6, 24, 96, c) A socce touamet has 64 paticipatig teams. I the fist oud, 32 games ae played. I each successive oud, the umbe of games deceases by oe half. Fid a ule fo the umbe of games played i the th oud, ad the total umbe of games played.

4 Algeba 2 AII.2 Geometic Sequeces ad Seies Notes Ms. Giese Page 4 Sums of Ifiite Geometic Seies Coside the seies: Is it a geometic seies? What is? Fid the fist 5 patial sums, S, S2, S3, S4, ad S5: o S = o S2 = o S3 = o S4 = = o S5 = Gaph these patial sums: What do you thik will happe as we icease? Examie the fomula fo the patial sum: S = a o What happes as gets vey big (appoaches ifiity)? Coside values of > < - - < < A ifiite geometic seies will covege if <; othewise it will divege Sum of a Ifiite Geometic Seies Fomula S = a, whe <

5 Algeba 2 AII.2 Geometic Sequeces ad Seies Notes Ms. Giese Page 5 Examples: Fid the sum, if possible a) i 5(0.8) i b) You Ty Fid the sum of the ifiite seies, if possible a) k 2 k b) j j c) Recusive Fomulas So fa, we have woked with explicit fomulas fo aithmetic ad geometic sequeces o The explicit ule fo the th tem of a aithmetic sequece: o The explicit ule fo the th tem of a geometic sequece: We ca also defie tems of a sequece ecusively o Recusive fomulas defie oe o moe iitial tems, ad the each futhe tem is defied as a fuctio of pecedig tems. o Examples of ecusio: Fiboacci sequece Iitial tems: a=0, a2= Recusive equatio: a = a- + a-2 Expad: Factoial fuctio Iitial tems: 0! = Recusive equatio:! = *(-)! (fo > 0) Expad:

6 Algeba 2 AII.2 Geometic Sequeces ad Seies Notes Ms. Giese Page 6 Recusive fomulas fo aithmetic ad geometic sequeces o Recusive fomula fo aithmetic sequeces: Add the commo diffeece to the pevious tem o Recusive fomula fo geometic sequeces: Multiply the commo atio to the pevious tem o Examples: Wite a ecusive ule fo the sequece a) 3, 3, 23, 33, 43, b) 6, 40, 00, 250, 625, c) Wite the fist 5 tems of the sequece: a =3; a =a o You Ty a) Wite the fist 6 tems of the sequece: a 0 =, a =a b) Wite the fist 6 tems of the sequece: a =, a =3a - c) Wite a ecusive ule fo the sequece: 2, 4, 98, 686, 4802, d) Wite a ecusive ule fo the sequece: 9, 3, 7,, -5, e) Wite a ecusive ule fo the sequece:,, 2, 3, 5, f) Wite a ecusive ule fo the sequece:,, 2, 6, 24,

7 Algeba 2 AII.2 Geometic Sequeces ad Seies Notes Ms. Giese Page 7 Fomula Summay: Sequeces Explicit Recusive Aithmetic a = a + ( )d a = a- + d Geometic a = a a = a- Seies Sum of fist iteges: i i = ( ) 2 Sum of fist 2 iteges: Sum of aithmetic seies: Sum of geometic seies: i i 2 = S = S = ( )(2 ) 6 a i i = ( a a 2 a i = i i ) a( ), Sum of ifiite geometic seies S = a a i = i i, <