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 to use subscript otatio rather tha the stadard fuctio otatio. For example, we could use a to represet the first term of the sequece, a2 to represet the secod term, a3 the third term ad so forth where a would represet the th term Note: Occasioally, it is coveiet to begi a sequece with somethig other tha a, such as a 0, i which case the sequece would be a 0, a, a 2, a 3,, a 2 a,, a is the term before a ad so forth) a, a, (ote how Ex. : List the first 5 terms of: Parts a, b, ad c represet explicitly defied sequeces. a) a 3 ( ) b 2 c) 2 c 2 d) The recursively defied sequece a a 5, a 25
AMHS Hoors Algebra 2 - Uit 6 Factorial The symbol! (read factorial ) is defied as! ( ) ( 2) 432 where 0!. 27 For example, 6! 65432 720 Ex. 2: Simplify the ratio of factorials: a) 25! 23! ( 2)!! c) (2 )! (2 2)! Patter Recogitio for sequeces Ex.3: Fid a explicitly defied sequece a whose first five terms are a) 2 4 8 6 32,,,,,... 3 5 7 9 2 8 26 80 242,,,,,... 2 6 24 20 c) 2 3 4 x x x, x,,,,... 2 6 24
AMHS Hoors Algebra 2 - Uit 6 28 Series Defiitio of a Series (Fiite ad Ifiite): Cosider the ifiite sequece a, a 2, a 3, a i,. The sum of the first terms of the sequece is called a fiite series or the partial sum of the sequece ad is deoted by a a2 a3... a ai where i is called the idex of summatio, is the upper limit of summatio ad is the lower limit of summatio i 2. The sum of all the terms of a ifiite sequece is called a ifiite series ad is deoted by a a2 a3... ai... ai i Ex.: Fid the sum: a) 5 i 3i 6 k3 2 ( k ) c) 8 ( ) 0!
AMHS Hoors Algebra 2 - Uit 6 d) 3 i 3 ( ) 0 i 29 Ex. 2: Use sigma otatio to write the sum 3() 3(2) 3(3) 3(9) Arithmetic Sequeces ad Partial Sums What behavior do the first two sequeces have that the third oe does ot?. 7,, 5, 9,. 2. 2, -3, -8, -3, -8 3., 4, 9, 6, A sequece is arithmetic whe the differece betwee cosecutive terms is costat. We call this differece the commo differece ( d ) where d a a. Ex.: Fid a formula for the th term of the arithmetic sequece 7,, 5, 9,. where a is the first term. The th term of a arithmetic sequece has the form a a ( ) d where d is the commo differece ad a is the first term. Ex.2: Fid a formula for the th term of the arithmetic sequece whose commo differece is 3 ad whose first term is 2. List the first five terms of this sequece.
AMHS Hoors Algebra 2 - Uit 6 30 Ex.3: The fourth term of a arithmetic sequece is 20, ad the 3 th term is 65. Write the first several terms of this sequece. Ex. 4: Fid the ith term of a arithmetic sequece whose first two terms are 2 ad 9. The sum of a fiite arithmetic sequece with terms is give by: S ( a a) 2 Note: Ex.5: Fid the sum: +3+5+7+9++3+5+7+9 Ex.6: Fid the 50 th partial sum of the sequece 5,6,27,38,49,. Ex. 7: 500 (2 8) Ex. 8: 50 0 (50 3 )
AMHS Hoors Algebra 2 - Uit 6 Geometric Sequeces ad Series 3 List the first five terms of the geometric sequece a 2 A sequece is geometric whe the ratios of cosecutive terms are costat. We call this costat the a commo ratio (r) where r. a Ex.: Determie if the followig sequeces are geometric ad if so, determie the commo ratio. a) 2, 36, 08, 324,,,,,,... 3 9 27 8 243 c), 4, 9, 6,..
AMHS Hoors Algebra 2 - Uit 6 32 The th term of a geometric sequece has the form a ar where r is the commo ratio ad a is the first term. Ex. 2: Write the first five terms ad the geeral term of the geometric sequece whose first term is a ad whose commo ratio is r 2. 3 Ex. 3: Fid the 5 th term of the geometric sequece whose first term is 20 ad whose commo ratio is.05. Ex. 4: Fid a formula for the th term of the followig geometric sequece. What is the ith term of the sequece? 5, 5, 45 Ex. 5: The fourth term of a geometric sequece is 25 ad the 0 th term is 25 64. Fid the 4th term (assume all terms are positive).
AMHS Hoors Algebra 2 - Uit 6 33 The th partial sum of a geometric sequece with commo ratio of r ad first term of a is give by Note: i r S ar a( ) r i Ex. 6: Fid the sum: a) 2 (4(.3) ) 7 0 2 The Sum of a Ifiite Geometric Series (or simply Geometric Series) If r, the the ifiite geometric series has the sum S a r i a ( r ) i0 If r, the the series does ot have a sum ad diverges to ifiity.
AMHS Hoors Algebra 2 - Uit 6 34 Ex. 7: Fid the sum: a) 0 (4(.6) ) 3 + 0.3 + 0.03 + 0.003 +
AMHS Hoors Algebra 2 - Uit 6