Sequences and series Mixed exercise 3

Transcription

1 eqees seies Mixe exeise Let = fist tem = ommo tio. tem = 7 = 7 () 6th tem = 8 = 8 () Eqtio () Eqtio (): m to te tems 0 o Diffeee betwee 0 = =.6 ( s.f.) 0 The ommo tio is. tem is 80 = 80 = 80 () b bstitte bk ito Eqtio (): The fist tem is th tem is. =. =. () Eqtio () Eqtio (): Hee ommo tio = 0.. m to ifiity b bstitte = 0. ito Eqtio (): ( 0.) 00 The fist tem i the seies is 00. m to ifiity is Peso Etio Lt 07. Copyig pemitte fo phsig istittio oly. This mteil is ot opyight fee.

2 ( ) m to tems o 00( 0. ) ( 0.) (to s.f.) Reqie iffeee ( s.f.) m to ifiity is Reple with Reple with b Reple with ( s.f.) Reple with Reple with 9 Reple with tems A geometi seies with = 76,. Use ( ) iseqivlet to Peso Etio Lt 07. Copyig pemitte fo phsig istittio oly. This mteil is ot opyight fee.

3 b geometi seieswith tems times whee b bstitte = 0.8 ito Eqtio (): The fist tem is 0. Use with = 0 = m to ifiity is 0. Use ( ) ( s.f.) ( ) 0( 0.8 ) = = 0.89 ( s.f.) Let = fist tem = the ommo tio of the seies. We e give tem = 6. = 6. () b log 0. log o the vle will be less th 000 fte 9.9 yes. th tem =. =. () Eqtio () Eqtio (): The ommo tio is 0.8. Peso Etio Lt 07. Copyig pemitte fo phsig istittio oly. This mteil is ot opyight fee.

4 7 p(q ) p(q ) p(q ) p(q ) (q ) (q )( q ) q 8q 6q q q 9q 0 ( q)(q) 0 q o q b q, 896, = 6p, = p p = ( 0.7 ) 0.7 = = + ( + ) + ( + ) ( + ( )) + (+( )) Tig seies o: = ( + ( )) + ( + ( )) +... ( + ) + Aig the two sms: = ( + ( )) + ( + ( )) +... ( + ( )) + ( + ( )) Thee e lots of ( + ( )): = ( + ( )) : b The fist 00 tl mbes e,,, Aithmeti seies with =, =. Usig with =, = gives () olve = 000: ( ) = 000 = = o.7 mst be positive, so =.87. If the sm hs to be gete th 000 the =. We ee to fi = This seies is ithmeti with =, =, = 00. Usig with =, = = 00 gives Peso Etio Lt 07. Copyig pemitte fo phsig istittio oly. This mteil is ot opyight fee.

5 0 Let = fist tem = ommo iffeee. m of the fist two tems = = 7 + = 7 0th tem = 6 Usig th tem = + ( ) + 9 = 6 (Note: + is ommo eo hee) O two simlteos eqtios e + = 7 () + 9 = 6 () + 8 = () (() ) 7 = 7 (() ()) = ( 7) bstitte = ito (): = 7 = 0 = Theefoe, fist tem = ommo iffeee = b sig bstittig =, = gives 60 = 0( + 9 ( )) = 0(0 77) = 0 ( 7) = 80 m of iteges ivisible by whih lie betwee 00 = This is ithmeti seies with =, = L = 99. Usig L Theefoe, thee e of these iteges p to 00. L b m of iteges ot ivisible by = ( 99 00) Aithmeti seies with =, L=00, =00 ( ) Fompt, this eqls ( 00) o sm of iteges ot ivisible by = = 67 Peso Etio Lt 07. Copyig pemitte fo phsig istittio oly. This mteil is ot opyight fee.

6 Let the shotest sie be x. 0 0 ( x x ) 67 ( x) 67 x 67 x Legth of shotest sie is m. Use th tem = + ( ): th tem is + = Use st tem =, tem = +, tem = + : sm of fist thee tems = m 8 8 st th This is ithmeti seies with =, = = = + = ( ) + = O simlteos eqtios e + = () : Usig (8 8 ) (8) () () Replig with U = k = = k Replig with = k = k = k(k ) = k k b bstitte = 6 k k = 6 k k 0 = 0 ( ) k k = 0 (ftoise) + = () () (): = ( ) = Commo iffeee = bstitte = bk ito (): + = = 6 Fist tem = 6 b th tem mst be gete th 8 + ( ) > ( ) > 8 (+6) ( ) > 88 ( ) ( ) > 7.6 (+) > 8.6 lest vle of = 9 (k )(k + ) = 0 k =, Peso Etio Lt 07. Copyig pemitte fo phsig istittio oly. This mteil is ot opyight fee. 6

7 6 We kow th tem = + ( ) th tem is k If the 7th tem is, the + 6 = + ( ) = k bstitte vles of : + = k We kow m to 6 tems is 7k + 9, theefoe k 9 ( + ) = 7k = 7k + 9 The simlteos eqtios e + = k () 6 + = 7k + 9 () () : + = k () () () : = 8k + 9 = 9 8k Fist tem is 9 8k. b bstittig this i () gives 9 8k k k9 k 9 Commo iffeee is k 9. k 9 8k 9 6 8k 9 k 9 8k9 k8 k 9 k k. Cllte vles of fist: 7 = 9 8k = 9 8. = 9 = k m to 0 tems is. p p p p p p o the seqee is peioi with oe. Peso Etio Lt 07. Copyig pemitte fo phsig istittio oly. This mteil is ot opyight fee. 7

8 7 b k 000 p p 00 p p k6 (k 6) 6 k 8 As the seqee is iesig: k k 6 k 8 k 6 b (k 8) 6 8k k k 6 k 8 8k k 66 (k ) Theefoe, 9 = 0, 60 b is ivisible by ( 0.8 ) =.69 (.p.) 0( 0.8 ) ( 0.8 ) log 0.8 log log log = 000, =.0 = = 6 00 b > > log.0 > log log log.0.00 Iitil ye ws 0, is itege, so (.0 ).0 = 7 = 7. e People my visit the oto moe feqetly th oe ye, some my ot visit t ll. It epes o thei stte of helth.,, 7, th tem = ( + ( )) = + b k + = 0 k = 0 q i q ( ( q )) p q( q ) p q q p q q p 0 Peso Etio Lt 07. Copyig pemitte fo phsig istittio oly. This mteil is ot opyight fee. 8

9 ii p > 0 q q p q q q q 0 q0 0 q0 0 ( q 8)( q 0) 0 q = 8 o 0 As q q0 0, q > 8 miimm mbes of ows is 9. =, 6.7 b ( )( + ) = 0 o As the seies is oveget, so = so = Chllege b 6 [ p( ) q( )] 6[ p( ) q( )] p ( ) q ( ) 6 p ( ) q ( ) 6 6 p p q q 9 p ( ) q( ) p q ( ) ( ) ( ) ( ) p q pq 9 pq olvig simlteosly: 0 6p q () 9p q () () (): p p q q Theefoe, ( ) ( ) = o it otis 8 igits. Peso Etio Lt 07. Copyig pemitte fo phsig istittio oly. This mteil is ot opyight fee. 9