means the first term, a2 means the term, etc. Infinite Sequences: follow the same pattern forever.

Transcription

1 9.4 Sequeces ad Seres Pre Calculus 9.4 SEQUENCES AND SERIES Learg Targets:. Wrte the terms of a explctly defed sequece.. Wrte the terms of a recursvely defed sequece. 3. Determe whether a sequece s arthmetc, geometrc or ether. 4. Fd (recursve ad explct) formulas for a arthmetc sequece. 5. Fd (recursve ad explct) formulas for a geometrc sequece. 6. Use summato otato. 7. Rewrte a seres usg summato otato. 8. Fd the sum of a arthmetc seres. 9. Fd the sum of a geometrc seres (both fte ad fte). 0. Tell whether a geometrc seres s coverget or dverget ad why. Day Day DAY : SEQUENCES Let s start wth some basc deftos Sequece: Ay set of umbers a specfc order. Each elemet the set s called a. We use a to detfy whch term we are talkg about. For example, a meas the frst term, a meas the term, etc. Two Types of Sequeces: o Ifte Sequeces: follow the same patter forever. o Fte Sequeces: follow the same patter for a fxed umber of terms. Explct Formula: Gves the sequece as a fucto of the umber of terms,. Recursve Formulas: Gves the frst term ad a fucto for fdg subsequet terms based o the prevous term. Always thk of as the prevous. Example : Tell whether gve formula s explct or recursve. The, fd the ext three terms of the sequece: a) a 0 b) a 57 a a 5for 9 -

2 9.4 Sequeces ad Seres Pre Calculus Arthmetc Sequeces: Ay sequece whose successve terms have a commo dfferece, d. Explct Formula: a a d( ) Recursve Formula: a # a a d for Example : Gve the arthmetc sequece,, 0,... a) Fd a recursve formula. b) Fd a 4. c) Fd a explct formula. Example 3: What s the st term of the arthmetc sequece, 4, 3,? Example 4: For the sequece below, wrte a explct formula ad fd the 0 th term. You do NOT have to fd a 3 the mddle terms! a a 4for Thk about ths Are the followg sequeces arthmetc? Fd a 0 for each of the sequeces WITHOUT fdg the mddle terms.. 79, 43, 8., 4, 6, 64, 9 -

3 9.4 Sequeces ad Seres Pre Calculus Geometrc Sequeces: Ay sequece whose successve terms have a commo rato, r. Explct Formula: a a r Recursve Formula: a # a a r for Example 5: Gve the sequece t 4 t 6( t ) for a) Fd the frst 4 terms of the sequece. b) Wrte a explct formula for the sequece. Example 6: Wrte a recursve formula for the sequece 50, 5,.5, 6.5,... Warm Up for 9.4 day : Fd the sum of the tegers from to 00 wthout a calculator. NO CHEATING!!!! 9-3

4 9.4 Sequeces ad Seres Pre Calculus DAY : SERIES Seres: the sum of the terms a sequece. Summato or Sgma Notato: Shorthad otato used to wrte fd the sum of. Most ofte used wth a explct formula. x x x x... x 3 Fte Seres: Seres wth a fxed umber of terms from a fte sequece. Ifte Seres: Seres whose terms cotue deftely; terms come from a fte sequece. Example 7: Let sequece A 8,0,3,4,7, 0 a) a 3 b) a 5 c). Fd: 3 a 6 d) a Example 8: Wrte a expresso for the sum of the fte seres usg summato otato. a) b) Arthmetc Seres: The sum of the terms a fte arthmetc sequece. a If you KNOW the last term a S Notce you eed to use ths formula. Example 9: Fd the sum of the frst 3 terms the arthmetc seres: Example 0: Fd the sum of the arthmetc sequece: 7, 0, 03,,

5 9.4 Sequeces ad Seres Pre Calculus Geometrc Seres: The sum of the terms a geometrc sequece. Sum of a fte geometrc seres ( a r S ) r Notce you eed to use ths formula. Example : Fd the sum of the frst terms of the geometrc sequece: 3 3 6, 3,,,... 4 Example : Fd the sum of the geometrc seres: Fdg the Sum of Ifte Geometrc Seres Whle we ca fd the sum of a fte geometrc or arthmetc seres, we ca oly fd a sum of a fte geometrc seres ad the oly whe a specfc codto s met. IF, THEN YOU CAN FIND THE SUM OF AN INFINITE GEOMETRIC SERIES. If the sum ca be foud, t s foud usg the formula. If a sum s possble, we say the seres. If a sum s ot possble, we say the seres. Example 3: Determe whether the seres coverges. If t coverges, gve the sum. a) j 3 j 4 b)

6 9. The Bomal Theorem Pre Calculus 9. THE BINOMIAL THEOREM Learg Targets:. Create Pascal s Tragle to at least sx rows usg patters.. Use combatos to fd the umbers ay row of Pascal s Tragle. 3. Expad bomal expressos. 4. Fd a specfc coeffcet of a term a bomal expaso wthout expadg the etre expresso. From the Explorato 9. ( ab) 0 ( ab) ( ab) 3 ( ab) ( ab) 4 5 ( ab) ab ab 0 0 ab ab ab 0 0 ab 3ab 3ab ab ab 4ab 6ab 4ab ab ab 5ab 0ab 0ab 5ab ab Pascal s Tragle If we just look at the tragular array of coeffcets the bomal expasos above, we get the frst 5 rows of the patter kow as Pascal s Tragle. Example : Fd the coeffcets the 6 th row of Pascal s Tragle. Patters of Bomal Expasos: ( a b ) The coeffcets for each term the expaso come from ether Pascal s Tragle or the Bomal Coeffcets: C r for r 0,,,3,.... The frst term s ab 0 or a. The last term s 0 ab or b. The expoets for a decrease by for each term whle the expoets for b crease by for each term. The sum of the expoets each term s. The expaso has + terms. Example : Use Pascal s Tragle to expad the expresso ( x ) 4 y. 9-6

7 9. The Bomal Theorem Pre Calculus Bomal Theorem: ( ab) C a Ca b C a b... C ab C b 0 or a b a a b a b b 0 r r r ( ) Example 3: Use the Bomal Theorem to expad the expresso ( x 3 ) 5 y. Example 4: Fd the coeffcet for the term cotag fdg the etre expaso. 7 y the expaso of (5x ) 9 y wthout actually 9-7