eqees seies Mixe exeise Let = fist tem = ommo tio. tem = 7 = 7 () 6th tem = 8 = 8 () Eqtio () Eqtio (): 8 7 8 7 8 7 m to te tems 0 o 0 0 60.7 60.7 79.089 Diffeee betwee 0 = 8. 79.089 =.6 ( s.f.) 0 The ommo tio is. tem is 80 = 80 = 80 () b bstitte 7 7 9 79 60.7 bk ito Eqtio (): The fist tem is 60.7. th tem is. =. =. () Eqtio () Eqtio ():. 80 0.06 0. Hee ommo tio = 0.. m to ifiity 60.7 60.7 8. b bstitte = 0. ito Eqtio (): 0. 80 ( 0.) 00 The fist tem i the seies is 00. m to ifiity is 8.. 00 00 0. 0.6 Peso Etio Lt 07. Copyig pemitte fo phsig istittio oly. This mteil is ot opyight fee.
( ) m to tems o 00( 0. ) ( 0.).8 76 66.6 67 (to s.f.) Reqie iffeee.8 0.000897 8.9 0 ( s.f.) 76 76 76 80 m to ifiity is 80. 9 Reple with Reple with b Reple with 9 0.876 ( s.f.) 9 76 9 60.8 Reple with Reple with 9 Reple with 8 7 9 76 60.8 9 tems A geometi seies with = 76,. Use ( ) 76 76 iseqivlet to Peso Etio Lt 07. Copyig pemitte fo phsig istittio oly. This mteil is ot opyight fee.
b geometi seieswith tems times whee b bstitte = 0.8 ito Eqtio (): 0.8 6. 6. 08. 0 The fist tem is 0. Use with = 0 = 0.8. 0 0 0 0.8 0. m to ifiity is 0. Use ( ).986 9.07 9.0( s.f.) 0.8 9.8 ( ) 0( 0.8 ) = 0 9.8... = 0.89 ( s.f.) Let = fist tem = the ommo tio of the seies. We e give tem = 6. = 6. () 6 0 000 0.8 887. b 0 0000.8 000 0.8 0. log 0. log 0.8 9.9 o the vle will be less th 000 fte 9.9 yes. th tem =. =. () Eqtio () Eqtio ():. 6. 0.8 The ommo tio is 0.8. Peso Etio Lt 07. Copyig pemitte fo phsig istittio oly. This mteil is ot opyight fee.
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, = 0.7 6 p 896 0.7 p = ( 0.7 ) 0.7 = 867.6 8 = + ( + ) + ( + ) +... + ( + ( )) + (+( )) Tig seies o: = ( + ( )) + ( + ( )) +... ( + ) + Aig the two sms: = ( + ( )) + ( + ( )) +... ( + ( )) + ( + ( )) Thee e lots of ( + ( )): = ( + ( )) : b The fist 00 tl mbes e,,,... 00. 00 00 00 99 0 0 00 9 + + 9 Aithmeti seies with =, =. Usig with =, = gives () olve = 000: ( ) = 000 = 000 000 = 0 000.87 o.7 mst be positive, so =.87. If the sm hs to be gete th 000 the =. We ee to fi = + + +... 99 + 00. 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.
0 Let = fist tem = ommo iffeee. m of the fist two tems = 7 + + = 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 =. 60 60 60 0 9 b sig bstittig =, = gives 60 = 0( + 9 ( )) = 0(0 77) = 0 ( 7) = 80 m of iteges ivisible by whih lie betwee 00 = + 6 + 9 + + + 99 This is ithmeti seies with =, = L = 99. Usig L 99 99 99 Theefoe, thee e of these iteges p to 00. L 99 0 6 7 b m of iteges ot ivisible by = + + + + 7 + 8 + 0 + + 00 ( 99 00) Aithmeti seies with =, L=00, =00 ( 6 9 99) Fompt, this eqls 6 7 00 ( 00) 000 80 00 o sm of iteges ot ivisible by = 80 00 6 7 = 67 Peso Etio Lt 07. Copyig pemitte fo phsig istittio oly. This mteil is ot opyight fee.
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 + ( ) > 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
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 6 6 7 k 9 ( + ) = 7k + 9 6 + = 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 9. 9 6. 9 7.. 0 0 0 0 9 0 9. 0 6 7. 0. 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
7 b 000 8 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 0 60 0 60 60 0 60 7 8 is ivisible by. 6 7 6 7 0 0 6.8 7 0( 0.8 ) 7 0.8 =.69 (.p.) 0( 0.8 ) 600 0.8 0( 0.8 ) 600 0. 0.8 0.8 log 0.8 log log log 0.8 0 = 000, =.0 = 000.0 = 6 00 b 000.0 > 0 000.0 > log.0 > log log log.0.00 Iitil ye ws 0, is itege, so 08. 8 8 000(.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
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 6.7 6.7 6.7 6.7 6.7 0 7 7 0 9 9 0 ( )( + ) = 0 o As the seies is oveget, so = so = 9 9 7 6.78 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, ( ) ( ) 00 00 00 =.6 0 7 o it otis 8 igits. Peso Etio Lt 07. Copyig pemitte fo phsig istittio oly. This mteil is ot opyight fee. 9