Sequences, Series, and the Binomial Formula

Transcription

1 CHAPTER Sequences, Series, nd the Binomil Formul. SEQUENCES. ; ; ; ; 6 ; 6 6. ðþ ; ðþ ; ð Þ 6; ðþ ; ðþ 6; 6 ð6þ. ðþ ; ðþ : ðþ ; ðþ ; ðþ ; 6 ðþ ; ; ; ; ; ; ; ; 8 ; 6 8 ; 6. 0; ; 6 ; ; ; 6 6. ; ; 6; 6 ; 0; ; 8; 8 ; ; ; 6 0. ; ; ; 6; 6 6;;6; 6 ð6;;6þ : ; ; ; 6;; 6; ; 6 6 ð6; Þ. ; ; ; ; 8 ; ; ; 0 ; 0; 0 ; 6 0. ft in:; ft ðþ in: ft in:; Since in: : ft; 6: 8: ft. 6000; ;00; n 6000 ðnþ00; 6000ð Þ $0;00; ð00þ $;00. ARITHMETIC AND GEOMETRIC SEQUENCES. Common difference, so rithmetic: d 8; ðþ Common rtio, so geometric: r 8 ; Geometric: r ; 6 ðþ ðþ 0. Arithmetic: d ; 8 ðþ. Neither common difference nor common rtio so it is neither rithmetic nor geometric. Arithmetic: d 0:; 0: 8ð0:Þ ð0:þ :6. Arithmetic: d 6 ; Geometric: r 0 or 0.; 6 ð0:þ 0:0000. Arithmetic: d ; ð ffiffi ffiffi Þ8ðÞ 8

2 88 CHAPTER SEQUENCES, SERIES, AND THE BINOMIAL FORMULA. Neither. It looks like n rithmetic sequence with lternting signs, so try ½: ðnþ:šðþ n ; 6 :. d 6 ; d; ; 60. r ; ; ;. 8 ðnþ; 6 ðn Þ; ðnþ; n 0. ;8 n; 6 ;6:66 n; 6 6 n ;6;6; ln 6 n ðnþln 6 ln ;6;6; ln ;6;6 n ln 6 8; n. ; ; ðnþd; 6d; 6d; d ; or :. 6 60d; 8 60d; : 60d; d 60 0: 8 mi. 0 ð0 Þd; 0 ð0:þ; 0 : :cm; 0 ðnþð0:þ; 0:ðnÞ; n 0; n. () Geometric sequence: Here r ð0:6þ 0:88 nd 0, so the concentrtion is reduced to 00 m when 00 0ð0:88Þ n. Thus, n 00 0 ð0:88þn nd so, ðn Þ ln ð0:88þ ln 0. Dividing, we get n ln 0 :68, nd hence n :68. Thus, lnð0:88þ the concentrtion will be reduced to 00 m bout ;68 mi from the first monitor or bout :68 mi from the sill. (b) Here n : 0ð0:88Þ n nd so n : ln 0 6: which is n : mi lnð0:88þ from the monitor so bout 8: mi from the sill SERIES P 0 k k P 0 n0 P k P 6 k PP ðþ n n n ðk Þ 0 0 k 6 6 n 8 6 n P 8 i i n ;; 6; :. Aritimetic: d ; 0 0; S 0 0ð0Þ 6. Geometric: r ; S 8 8. Arithmetic: d ; 6 8; S ð68þ. Arithmetic: d 0:; 0 0: ð:þ :; S 0 0 ð0: :Þ :. Geometric: r ; S 6 ð Þ 6 ð Þ 6 0:6. Arithmetic: d :; 0: ð:þ 6; S ð0: 6Þ :8. Geometric: r ffiffiffi ; S ffiffiffi ffiffi ffiffi ;0:. Geometric: r ; S First time it hits the ground it hs only fllen 80m. After this it rises nd flls so 80ð:8Þ. So this is sequence 8060ð:8Þ60ð0:8Þ 60ð0:8Þ 60 60ð0:8Þ 60ð0:8Þ 60ð0:8Þ :8 0:8 80 :m when it hits the fourth time. 60 0:80 0:8 80 6:0 when it hits the tenth time.. In 6 ft sn there will be studs 6 00 rt. The longest is8 0 nd the shortest is 8 00 or ft. S 8 68 ft. This is only hlf of the roof. The whole roof will be twice tht or 0 ft.. The rte of 8% comounded monthly is 8 % er month 0:006 er month, yers hve 6 months A 000ð 0:006Þ 6 $0:. This is $. more thn exercise #.

3 SECTION. 8. () Here the sequence is $0;000 $0;000 $0;000 $0;000 $6; 000. Here 0;000 is the cost of the nd floor, so 6;000 nd S ð0;0006;000þ ;;000 is the cost of floors 0. So, the cost of 0-story building is $0;000 $0;000 $;;000 $;66;000. (b) $0;000 $0;000 S $;68;000: ðs cost of floors 0 ð0;000 6;000Þ 6;6;000:Þ. INFINITE GEOMETRIC SERIES. r so converges; S. r so converges; S or 0:6. r so converges; S. r 0 so converges; S : r 0: so converges; S 0:0 0 0:0.... r ð0:8þ so converges; S 0:.... r so diverges. r 0 so diverges 0. r so converges; S 0 0: 0:0 0: 0 0:8 : r ffiffi ffiffi. r ffiffi ffiffi ffiffi so converges; S ffiffi ffiffi ffiffi ffiffi : so diverges. 0:; r 0:; S 0: 0: 0: 0:. 0:; r 0:0; S 0:0 0: 0:. :; r 0:000; S : 0 0:000 : 0:. 6: : ð0:00 0:00000 Þ; 0:00; r 0:00; S 0:00 : 0 0 ; 6: After the first bounce it is geometric series with 8; r 0:8; S 8 0:8 8 0: 0. The totl distnce is 0 m.. Geometric series with 0; r 8 0 0:8 so S 0: : 0 rev.. THE BINOMIAL THEOREM. ð Þ 6 6. ðx Þ ðxþ ðxþ ðþ6ðxþ ðþ ðxþðþ ðþ 8x 08x x x x. d 6 x 6 6 x d x d d 0 x d x 6 x d d 6 x6 6 x d 6 x d 6 x d x d xd d 6. b b 0 b 0 b b b b 0 b 60 b 60 b 0 b. ð bþ 0 6 b b b b b 6 b 6 b 6 b b b b b b 6 b. ðt Þ t 8 8 t ðþ 8 t 6 ðþ 8 t ð Þ 8 t ðþ 8 t ðþ 8 6 t ðþ 6 8 tðþ 8 8 ðþ 8 t 8 8t 8t 6 6t 0t 6t 8t 6 8t 8. ð Þ ðþ 6 6 ðþ ðþ 6 ðþ ðþ 6 ðþ ðþ 6 ðþ ðþ 6 ðþðþ 6 6 ðþ

4 0 CHAPTER SEQUENCES, SERIES, AND THE BINOMIAL FORMULA. x y 0 ð x yþ ð x yþ 6 x ð y Þ ð x yþ ð x yþ ð x yþ 6 ð x yþ x y 6 x y 6 x0 y 8 x8 y 6 x6 y x y 6 x y 6 8. ðx yþ 0 x x y x 0 y x y x x y 66x 0 y 0x y. ð Þ ðþðþ ðþðþ ðþ! ðþðþðþ ðþ!. ð bþ ð b Þð Þ b! ð Þð Þð Þ b b! b b 8. ð:þ ð0:þ ð0:þ! ð0:þ! ð0:þ 0: 0:06 0:00 :6 ffiffiffiffiffiffi. : ð0:þ ð0:þ ð Þ ð0:þ! ð Þð Þ ð0:þ 0:0 0:00 0:00006! :0. :0 ð0:0þ ð0:0þ ð Þ! ð0:0þ 0:008 0:0008 :008. 0: ð0:0þ ð0:0þ 0:0 0:8. Counting from 0, the sixth term is ð Þðx Þ 0 y 00x 0 y. Fifth term is ð ÞðxÞ8 ðyþ 6;0x 8 y. Term involving b is the th term: ð Þ0 b 00 0 b. () v v c ð Þð Þ! v c v (b) mc mv mv 8c c v 8c =. ðr Þ = r " r r #! r r r 8r r r 8r c CHAPTER REVIEW. ; ; ; 6 ; ; 6 8. ; 0 ; ðþ ; ðþ 6 ; ðþ ; 6ðÞ 8. ; ; 0 ; ; 6 ;. ; ; ; ; ; 6 8. Arithmetic: d ; 0 0 ðþ. Arithmetic: d ; 6ðÞ. Geometric: r 6 ; 0 6 ;; :68 0 0: d 8 6; 6d; 6 6; 6. () 8 8 (b). 8. Arithmetic: d ; 0 ; S 0 ð Þ ðþ 6 0. Geometric: r ; S. Converges: r ; S. Diverges: r 0. Converges: r ffiffi ; S ffiffi ffiffi :660 ffiffi : ffiffi ffiffi ffiffi ffiffi. 0:8; r 0:00; S 0:8 0: 8. ð Þ

5 CHAPTER TEST x 6 x 66 x ðy ðy. y Þ x Þ ðy 0 x Þ ðy x Þ 6 x ðy Þ ðy Þ 6 x6 6 6 x y 6 x y x y 6 x y 8 xy 0 y. ðx yþ ðxþ ðxþ y ðxþ y ðxþ y ;68x ;60x y 860;60x y ;86;680x y. ð xþ ðxþ ðþð6þ! x ðþð6þðþ! ðxþ x x x 0. 6 x ðyþ 6 ;80;60x y 6. :0 ð0:0þ ð0:0þ ð Þ! ð0:0þ 0:00 0:0000 :00. % er month 0:00; 0 yers 0 months; A 00ð:00Þ 0 $:6. S 6 0 ð Þ6 6:6 cm. First convert 0 ft to inches. Since 0 000, we see tht 0 ft 000 in. S n n ð n Þ; n ðnþd so n 0 ðnþ n. Substituting, we get S n n ð0 nþ n nn ð nþ 000. Hence, n n Dividing by, we hve n n The qudrtic formul gives nswers of 0:6 nd :8. Since n must be ositive, the nswer is.8 s CHAPTER TEST. ; ; ; ; ;. Arithmetic: d :; 0 ð:þ :. P k ðk Þ. 0:; r 0:00; S 0: 0:. Arithmetic Series: 000; d 00; 000 ð00þ8;00; S ð000 8;00Þ ;000 coies