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AYT Matematik Ders İşleyiş Modülleri 4. Modül Diziler

Published by Nesibe Aydın Eğitim Kurumları, 2019-08-24 01:31:33

Description: AYT Matematik Ders İşleyiş Modülleri 4. Modül Diziler

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www.aydinyayinlari.com.tr ·/÷7&34÷5&:&)\";*3-*, ÜNİVERSİTEYE HAZIRLIK 4. MODÜL MATEMATİK - 2 DİZİLER ³ Sayı Örüntüleri t 1 ³ Dizi t 3 ³ Aritmetik Dizi t 15 ³ Geometrik Dizi t 24 ³ Karma Testler t 35 ³ Yeni Nesil Sorular t 40 1

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·/÷7&34÷5&:&)\";*3-*, 4. MODÜL %÷;÷-&3 www.aydinyayinlari.com.tr ÖRNEK 7 ÖRNEK 9 \"õBóEBLJGPOLTJZPOMBSOO`N+J¿JO CJSHFS¿FLTBZEJ- (FOFMUFSJNMFSJBöBôEBWFSJMFOEJ[JMFSJOJMLÑÀUFSJN- [JTJPMVQPMNBEóOHËTUFSJOJ[ MFSJOJOUPQMBNOCVMVOV[ %J[JPMBOMBSOJMLEÌSUUFSJNJOJCVMVOV[ a) (an) = f 1 p n b ) f (n) = n a ) G O =O n+3 b ) CO =++++ O- c ) f (n) = 3n E G O =MPHO c ) DO =+ + + ++O n-2 f ) f (n) = n2 - 16 a1 + a2 + a3 = 1 1 1 11 e ) f (n) = n + 3 a) ++= 123 6 b) b1 + b2 + b3 = 1 + (1 + 3) + (1 + 3 + 5) = 14 H f 1 , 2 , . . . , n , . . . p I c) c1 + c2 + c3 = 2 + (2 + 22) + (2 + 22 + 23) = 22 47 3n + 1 (an) = f n-2 p n+2 B %J[JEJSG = 1, f ( 2 ) = 2, f ( 3 ) = 3, f ( 4 ) = 4 ÖRNEK 10 1234 (FOFMUFSJNJ C %J[JEJSG = , f ( 2 ) = , f ( 3 ) = , f ( 4 ) = 4567 Z 3n + 1, n tek ]] D %J[JEFôJMEJSO=JÀJOSFFMTBZCFMJSUNF[ an = [ 2, E %J[JEJS ]] 2n + 1 n çift \\ f ( 1 ) = 0, f ( 2 ) =MPH G =MPH G =MPH olan ( an EJ[JTJJÀJOB1 + a2 + a3 toplaNLBçUS F %J[JEJS = 4, = 2 = 10 f ( 1 ) = 2, f ( 2 ) = 5 , f ( 3 ) = 6 , f ( 4 ) = 7 a a , a 5 G %J[JEFôJMEJSO= WFJÀJOSFFMTBZCFMJSUNF[ 1 2 3 12 3 4 2 72 H %J[JEJS , , , 14 + = 4 7 10 13 55 I %J[JEFôJMEJS HFOFMUFSJNJCFMJSMJEFôJMEJS –1 1 2 %J[JEJSB1 = 3 , a2 = 0, a3 = 5 , a4 = 6 ÖRNEK 8 ÖRNEK 11 (an) = f 2n + 1 p (FOFMUFSJNJBnPMBOCJSEJ[JEFB1 =WFrn > 1 için, 3n - 1 aO =O-+ aO- EJ[JTJOJOLBÀODUFSJNJ 3 UÑS PMEVôVOBHÌSF B12LBÀUS 4 a - an-1 = 3n - 1 n a2 - a1 = 3 · 2 - 1 = 5 2n + 1 3 a3 - a2 = 3 · 3 - 1 = 8 = a12 - a11 = 3 · 12 - 1 = 35 3n - 1 4 8n + 4 = 9n - 3 7 = n \"UFSJN a - a = 220 j a = 221 12 1 12 7.D GWFIEJ[JEFôJMEJSB C b) 1/4, 2/5, 3/6, 4/7 4 11 72 11. 221 E MPH MPH MPHF 5 , 6, 7 H 9. a) , b) 14, c) 22 10. m 8. UFSJN 65

www.aydinyayinlari.com.tr %÷;÷-&3 4. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 12 ÖRNEK 15 (an) = f 7n + 12 p (an) = f 2n - 8 p n n+3 EJ[JTJOJOLBÀUFSJNJUBNTBZES EJ[JTJOJOLBÀUFSJNJOFHBUJGSFFMTBZES 7n + 12 12 –3 4 = 7 + PMVQO= EFôFSMFSJJÀJO +– + nn UFSJNJUBNTBZES n ` N+PMBDBôOEBOO= PMVQUFSJNJOFHBUJGUJS ÖRNEK 13 ÖRNEK 16 (FOFMUFSJNJ (an) = f 6n + 10 p an = 2n + 1 n+1 2n EJ[JTJOJOUBNTBZUFSJNMFSJOJOUPQMBNLBÀUS PMBOCJSEJ[JOJOLBÀUFSJNJ 23 EFOCÑZÑLUÑS 22 (a ) = 6n + 10 =6+ 4 j n =WFO= 3 için tam n n+1 n+1 TBZESB1 + a3 = 8 + 7 =PMVS 2n + 1 23 > 2n 22 44n + 22 > 46n 22 > 2n 11 > n j 10, 9, ... , 1 UFSJN ÖRNEK 14 (an) = f n2 - 16 p ÖRNEK 17 (7 - 2n) . (3n - 4) BO = O -O+ EJ[JTJOJOLBÀUFS JNJQP[JUJGSFFMTBZES EJ[JTJOJOFOLÑÀÑLUFSJNJLBÀUS -4 4/3 7/2 4 a = n2 - 9n +QBSBCPMEFOLMFNJPMEVôVOEBO QB- - +-+ - n SBCPMÑOUFQFOPLUBTOOBQTJTJOEFFOLÑÀÑLUFSJNJBSB- OS5FQFOPLUBTOOBQTJTJ -b 9 = = 4.5 PMEVôVOEB n ! N+ PMBDBôOEBO O = JÀJO TBôMBOS UBOF UFSJNJ 2a 2 QP[JUJGUJS n =WFO=FCBLMNBMESB4 = a5 =ES 12. 6 13. 15 14. 1 5 15. 3 16. 10 17. 0

·/÷7&34÷5&:&)\";*3-*, 4. MODÜL %÷;÷-&3 www.aydinyayinlari.com.tr ÖRNEK 18 ÖRNEK 21 BO = -O +O+ (an) = f 3n + 2 p EJ[JTJOJOFOCÑZÑLUFSJNJLBÀUS 8 - kn b -5 EJ[JTJOJOTBCJUEJ[JPMNBTJÀJO L`3LBÀPMNBMES 1BSBCPMÑO UFQF OPLUBTOO BQTJTJ - = = 2.5 2a - 2 PMEVôVOEBOO= 2 WFO=FCBLMS 3BTZPOFM EVSVNEBLJ JGBEFMFSJ JÀFSFO CJS EJ[JOJO TBCJU EJ[J PMNBT JÀJO QBZ WF QBZEBTOO CJSCJSJOJO UBN LB- a2 = a3 =PMVS UPMNBTHFSFLJS 32 = & k = -12 –k 8 Sabit Dizi Sonlu Dizi TANIM #ÐUÐO UFSJNMFSJ CJSCJSJOF FõJU PMBO EJ[JZF sabit TANIM EJ[JEFOJS BO EJ[JTJTBCJUEJ[JJTF L` Z+ ve AL = { L} a = a== aO PMVS AL e Z+PMNBLÐ[FSF UBONLÐNFTJ\"LPMBOIFS GPOLTJZPOBTPOMVEJ[JEFOJS ÖRNEK 19 \"LTJ CFMJSUJMNFEJóJ TÐSFDF EJ[J TË[ÐOEFO son- TV[EJ[JBOMBõS \"öBôEBLJMFSEFOIBOHJMFSJTBCJUEJ[JEJS a ) BO = b ) BO = TJOOÕ ÖRNEK 22 c ) BO = DPTOÕ E BO = - O+ A Z3 (an) = f n+1 p n+2 a) ( a ) = TBCJUEJ[JEJS n b) ( a ) = TBCJUEJ[JEJS TPOMVEJ[JTJOJOUFSJNMFS JOJZB[O[ n c) ( an ) = ( -1, 1, -, .... , ( -1 )n TBCJUEJ[JEFôJMEJS E Bn ) = ( -1, -1, -1, .... ( -1 )2n+1 TBCJUEJ[JEJS A5 j an EJ[JTJOJOJMLUFSJNJOJJÀFSNFLUFEJS 23456 *,,,, 4 34567 ÖRNEK 20 %m/*m (an) = k - 3 n2 + 9 - k2 n + k + 2 + r + r ++ rO- = 1- rn EJS EJ[JTJOJOTBCJUEJ[JPMNBTJÀJOL` N kaç olmaMES 1- r an TBCJU EJ[J PMNBT JÀJO O ZF CBôM PMNBNBT HFSFLJS #VZÑ[EFOOMJJGBEFMFSJOLBUTBZMBSPMNBMES 18. 12 19. B TBCJUC TBCJUD TBCJUEFôJME TBCJU20. 3 6 23456 21. –12 22. * , , , , 4 34567

www.aydinyayinlari.com.tr %÷;÷-&3 4. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 23 ÖRNEK 26 A8 = { w }PMNBLÐ[FSF aO : A8 Z3 BO = O EJ[JTJWFSJMJZPS ^anh = f n2 + 3n + 2 p ve CO =$ O+ 2 #VOBHÌSF( an EJ[JTJOJOCÑUÑOUFSJNMFSJOJOUPQMB- NOCVMVOV[ EJ[JMFSJOJOFöJUPMVQPMNBEôOCFMJSMFZJOJ[ a1 = 21, a2 = 22, ..., a8 = 28 an = ^ n + 2 h·^ n + 1 h WFCn = ^ n + 1 h.n 2 + 22 + .... + 28 = 2 · ( 1 + 2 + .... + 27 ) 2 2 = 2 · ( 28 - 1 ) = ( 29 - 2 ) = 510 EJ[JMFSJ IJÀCJS O ` N+ JÀJO CJSCJSJOF FöJU PMNBEôOEBO FöJUEJ[JMFSEFôJMEJS ÖRNEK 27 ÖRNEK 24 BO = f sin (n + 1) π pWF CO = - O 2 ÷ML UFSJNJ PMBO WF BSU BSEB HFMFO UFSJNMFSJ öFS B[BMUMBSBL FMEF FEJMFO EPôBM TBZ EJ[JTJ LBÀ UFSJNMJ- EJ[JMFSJFöJUNJE JS EJS a1 = 24 ^ n + 1 hr ( an ) = ( 0, -1, 0, 1, ..., sin 2 , ...) a = 24 - 2 · 1 = 22 ( bn ) = ( -1, 1, -1, ..., (-1)n, ...) 2 rn ! N+ anâCnEJS a13 = 24 - 2 · 12 = 0 &UFSJNMJEJS &öJU%J[JMFS ÷OEJSHFNFMJ%J[J TANIM TANIM rO` Z+J¿JOBO =COPMVZPSTB #JSUFSJNJLFOEJOEFOËODFLJCJSWFZBCJSLB¿UF- BO WF CO EJ[JMFSJOFFöJUEJ[JMFSEFOJSWF SJNDJOTJOEFOJGBEFFEJMFCJMFOEJ[JMFSFJOEJSHF- BO = CO õFLMJOEFHËTUFSJMJS NFMJ EJ[J UBONMBNB CBóOUTOB EB JOEJSHF- NFCBôOUTEFOJS öLJEJ[JOJOFõJUPMNBEóOHËTUFSNFLJ¿JOFOB[ CJSFSUFSJNJOJOGBSLMPMEVóVOVHËTUFSNFLZFUFS- ÖRNEK 28 MJEJS ÷OEJSHFNFCBôOUT ÖRNEK 25 aO+ = aO +O BO = O- WF CO = _ n2 i olan anEJ[JTJJÀJOB1 =PMEVôVOBHÌSF B12LBÀUS EJ[JMFSJOJOFöJUPMVQPMNBEôOCFMJSMFZJOJ[ a2 = a1 + 2.1 n2 = 3n - 2 a3 = a2 + 2.2 n2 - 3n + 2 = 0 (n - 2) (n - 1) = 0 + a12 = a11 + 2.11 n =WFO= 1 #VEVSVNEBBnWFCnTBEFDFO=WFO=JÀJOFöJU a = a + 2.(1 + ... + 11) PMEVôVOEBOFöJUEJ[JEFôJMEJSMFS 12 1 = 4 + 2· 11.12 = 136 PMVS 2 23. 510 24. 13 25. &öJUEFôJM 7 26. &öJUEFôJM 27. &öJUEFôJM 28. 136

·/÷7&34÷5&:&)\";*3-*, 4. MODÜL %÷;÷-&3 www.aydinyayinlari.com.tr ÖRNEK 29 ÖRNEK 31 BO = (FOFMUFSJNJBnPMBOCJSEJ[JEFB1 =WFrn ` N+ için, EJ[JTJOJOHFOFMUFSJNJOJWFJOEJSHFNFCBôOUTOCV- aO+ = O+ BO MVOV[ PMEVôVOBHÌSF CVEJ[JOJOHFOFMUFSJNJOJCVMVOV[ r B1= 1 a aa a a1 = 1 n+1 =n+1 j 23 n = 2, = 3, = - 1 Z a a a =n a ( 2 1 ) · a n 12 a 2 1 n–1 aa a a3 = ( 3 - 1 ) · 1 Z a2 23 n j · . . . = 2.3 . . . n aa a a4 = ( 4 - 1 ) · 2 Z a3 12 n–1 a5 = ( 5 - 1 ) · 6 Z a4 a n j = 2· . . . . · n = n!PMVS 1 h an = (n - 1) an - 1EJS a ÖRNEK 32 n O`/WFOãPMNBLÐ[FSF a = n - 1PMVQ, a = BO = aOm +O JOEJSHFNFCBôOUTZMBWFSJMFOEJ[JOJOHFOFMUFSJNJOJ n–1 CVMVOV[ aa a 23 n = 1.2. . . . ^ n - 1 h · ... aa a 12 n–1 j an = (n - 1)! rn ` N+PMNBLÑ[FSFUBONMBOS a - an-1 = 3n n - a = 3 · 2 a - 1 = 3 2 a · 3 a 2 3 ÖRNEK 30 an - an-1 = 3 · n O`/WFOãPMNBLÐ[FSF an - 1 = 3 · ( 2 + .... + n ) a = an = an –1 an = 3n2 + 3n - 4 1 + an –1 2 JOEJSHFNFCBôOUTZMBWFSJMFOEJ[JOJOHFOFMUFSJNJOJ ÖRNEK 33 CVMVOV[ (FOFMUFSJNJBnPMBOEJ[JEFB1 = 1 , n `/WFO> 1 1 1+a 1 rn > 1 için, n–1 aO =O+ aO - = = +1 aa a PMEVôVOBHÌSF B20LBÀUS n n–1 n–1 a2 - a1 = 2 11 a3 - a2 = 3 - =1 aa n n–1 11 - =1 aa 21 11 - =1 aa 32 11 a - a = 20 - =1 20 19 aa a20 - a1 = 2 + 3 + ... + 20 n n–1 1 - 1= ^ n - 1 h1& 1 =n&a 1 = a a nn 20 · 21 a20 = 1 + 2 + 3 + ... + 20 = = 210 nn 2 29. ÷#Bn = ( n –1) · an–1 (5Bn = (n – 1)! 30. an = 1 8 31. n! 3n2 + 3n – 4 33. 210 n 32. 2

www.aydinyayinlari.com.tr %÷;÷-&3 4. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 34 'JCPOBDDJ%J[JTJ #JS Bn EJ[JTJOEFB1= -WFO>PMNBLÑ[FSF 7$1,0%m/*m BO = - O aOm öML JLJ UFSJNJ WF CVOEBO TPOSBLJ IFS UFSJNJ JOEJSHFNFCBôOUTZMBWFSJMFOEJ[JOJOHFOFMUFSJNJOJ LFOEJOEFO ËODFLJ JLJ UFSJNJO UPQMBN PMBO EJ[J- CVMVOV[. ZFGJCPOBDDJEJ[JTJEFOJS 3a = (-1)2. a 'JCPOBDDJEJ[JTJF = F =PMNBLÐ[FSF FO = FO- + FO- 2 1 O> O` Z JOEJSHFNFCBóOUTJMFUBONMB- 3a = (-1)3. a OBCJMJS 3 2 #VOBHËSF'JCPOBDDJEJ[JTJ 'O = CJ¿JNJOEFEJS 3an = (-1)n . an-1 &öJUMJLMFSUBSBGUBSBGBÀBSQMSTB 3n-1 . an = (-1)2+...+n . (-3) 3n-1 . an = (-1)1+ 2+...+n . 3 n^ n + 1 h a = 32–n.^ - 1 h 2 CVMVOVS n ÖRNEK 35 ÖRNEK 36 1- 5 n GO = 'JCPOBDDJEJ[JTJOJOHFOFMUFSJNJOEFLJf p JGB- GJCPOBDDJEJ[JTJWFO> O`;PMNBLÐ[FSF 2 EFTJOCÐZÐEÐL¿FBZBLMBõS a = B = BO = aO-BO- JOEJSHFNFCBóOUTWFSJMJZPS #VEVSVNEBGJCPOBDDJEJ[JTJOJOOUFSJNJ ( an JOHFOFMUFSJNJOJ Gn DJOTJOEFOCVMVOV[ f 'n = 1 f 1+ 5n p 52 EFóFSJOFFOZBLOEPóBMTBZES O fnl &OZBLOEPóBMTBZ a1 = 1 · 2 a2 = 1 · 3 a3 = a1 · a2 = 21 · 31 a4 = a2 · a3 = 3 · 21 · 31 = 21 · 32 a5 = a3 · a4 = 21 · 32 · 21 · 31= 22 · 33 :VLBSEBLJ GPSNÑM ZBSENZMB GJCPOBDDJ EJ[JTJOJO a = f . f PMVS UFSJNJOJIFTBQNBLJOFTJOJLVMMBOBSBLCVMVOV[ n 2 n–2 3 n–1 5 . 2, 24BMO[ 6 1 · f 1 + 5 p . 8.07 f6 = 52 n^ n + 1 h 9 ff 34. 2–n .^ - 1 h 2 35. 8,07 36. a = 2 n–2 . 3 n–1 3 n

TEST - 1 4BZ²SÑOUÑMFSJWF(FSÀFL4BZ%J[JMFSJ 1. \"öBôEBLJ GPOLTJZPOMBSEBO LBÀ UBOFTJ O ` N+ 4. a = O`/WFO>J¿JOBO =+ aO- JÀJOCJSSFFMTBZEJ[JTJEJS * G O =O- ** f (n) = 1 - n CJÀJNJOEFUBONMBOBOEJ[JOJOHFOFMUFSJNJBöB- n+2 ôEBLJMFSEFOIBOHJTJEJS ***f (n) = 3 *7 f (n) = 3n - 5 \" O- # O-$ O- n-4 % O- & O- 7 f (n) = 6 - n2 3 n2 - 16 7* f (n) = \" # $ % & 5. 3n + 9 fp n+1 EJ[JTJOJOUBNTBZUFSJNMFSJOJOUPQMBNLBÀUS 2. \" # $ % & 6. (an) = f xn + 2 p ve (bn) = f y-n pPMNBLÐ[FSF 3n - 1 3n - 1 &õLFOBS Ð¿HFO GBZBOTMBS JMF Ð¿HFO õFLMJOEFLJ CJS rn ` N+ için an = bnPMEVôVOBHÌSF YZÀBS- PEBOO [FNJOJ EËõFONJõUJS ¥FWSFTJ N PMBO CV QNLBÀUS PEBJ¿JOUBOFCFZB[WFUBOFTJZBIGBZBOTEË- õFONJõUJS \" - # - $ % & #VOB HÌSF ÀFWSFTJ N PMBO CJS PEB JÀJO LBÀ CFZB[WFLBÀTJZBIGBZBOTEÌöFOJS #FZB[ 4JZBI 7. BO = O -O+ \" # EJ[JTJOJOLBÀODUFSJNJOJOEFôFSJFOLÑÀÑLUÑS $ % \" # $ % & & 3. n+1 8. (an) = f n-3 p 2n - 7 n2 - 19 (an) = f p EJ[JTJOJOLBÀODUFSJNJFFöJUUJS EJ[JTJOJO SFFM UFSJNMFSJOEFO LBÀ UBOFTJ QP[JUJG EFôJMEJS \" # $ % & \" # $ % & 1. C 2. C 3. D 10 4. B 5. & 6. A 7. & 8. D

4BZ²SÑOUÑMFSJWF(FSÀFL4BZ%J[JMFSJ TEST - 2 1. (FOFMUFSJNJ an = 1 PMBOCJSEJ[JOJOLBÀUFSJNJ 5. aO+ = aO +O- n JOEJSHFNFCBôOUTZMBWFSJMFOCJSEJ[JEFB1 = 4 3 UFOCÑZÑLUÑS PMEVôVOBHÌSF B17LBÀUS 25 \" # $ % & \" # $ % & 2. (bn) = f 3n - c p n + 12 6. EJ[JTJOJOTBCJUEJ[JPMNBTJÀJOD`3OFPMNBM- ES \" - # - $ % & 3. B C D E #JS LBSFOJO J¿FSJTJOF LBSFMFS ¿J[JMJQ ¿J[JMFO LBSFMF- SJOLËõFHFOMFSJOJOBZSEóCJSFSCËMHFUBSBOBDBLUS :VLBSEBWFSJMFOTBZMBS'JCPOBDDJEJ[JTJOJOBSEõL ôFLJMEFJ¿J¿FUBOFLBSF¿J[JMNJõWFPMVõBOCËM- HFUBSBONõWFCËMHFUBSBONBNõUS UFSJNMFSJEJS #VOB HÌSF BZO öFLJMEF JÀ JÀF ÀJ[JMNJö LBSFMF- #VOBHÌSF a + d + 3c PSBOLBÀUS SJO BZSEô CÌMHF UBSBONBNöTB LBÀ CÌMHF d-b UBSBONöUS \" # $ % & \" # $ % 5 & 3 22 4. (an) = f 2n pEJ[JTJJÀJO 7. a =WFrO` N+PMNBLÐ[FSF (n + 2) ! an + 1 = 1 aO+ -= aO +O an 5 JOEJSHFNFCBôOUTZMBWFSJMFOEJ[JOJOHFOFMUFSJ- NJBöBôEBLJMFSEFOIBOHJTJEJS PMEVôVOBHÌSF OLBÀUS \" O +O- # O +O \" # $ % & $ O -O % O +O- & O +O+ 1. A 2. A 3. A 4. C 11 5. B 6. C 7. A

TEST - 3 4BZ²SÑOUÑMFSJWF(FSÀFL4BZ%J[JMFSJ 1. \"öBôEBLJMFSEFOLBÀUBOFTJCJSHFSÀFLTBZEJ[J- 5. (an) = f n2 + 3n + 24 p n+3 si EFôJMEJS *^ an h = f n+1 p **^ an h = f n–1 p EJ[JTJOJOUFSJNMFSJOEFOLBÀUBOFTJUBNTBZES 2n – 3 4–n ***^ an h = ^ n – 3 h *7^ an h = f n + 2 p \" # $ % & 3n + 3 7^ an h = a logn^ n + 1 h k7*^ an h = f 1 p 3n –1 \" # $ % & 6. BO = O -O+ EJ[JTJOJOLBÀUFSJNJEBOLÑÀÑLUÑS \" # $ % & 2. ^anh = f 4n – 3 p 3n + 4 EJ[JTJOJOLBÀODUFSJNJ 1 EJS 2 \" # $ % & 7. (an) = f n2 – 8n + 15 p 2n - 9 2n + 1 EJ[JTJOJOLBÀUFSJNJOFHBUJGUJS 5n + k 3. (an) = f p \" # $ % & EJ[JTJTBCJUCJSEJ[JPMEVôVOBHÌSF k . a5ÀBSQ- NLBÀUS \" # 4 $ 2 % 4 & 5 25 5 52 8. ( an EJ[JTJOEFB1 =WFrn ` N+ için, aO+ = aO + 4. BO = -O PMEVôVOBHÌSF B15LBÀUS % & \" # $ EJ[JTJOJOLBÀUFSJNJEPôBMTBZES \" # $ % & 1. C 2. A 3. A 4. A 12 5. B 6. B 7. A 8. D

4BZ²SÑOUÑMFSJWF(FSÀFL4BZ%J[JMFSJ TEST - 4 1. (an) = f 3n - 2 p 5. (FOFMUFSJNJBn = 3n + 1( n + PMBOCJSEJ[JEF 2n + 3 an + 2 = 36 EJ[JTJOJOLBÀUFSJNJ 3 UFOLÑÀÑLUÑS an + 1 4 PMEVôVOBHÌSF OLBÀUS \" # $ % & \" # $ % & 2. Y-Z Y+Z EJ[JTJ GJCPOBDDJ EJ[JTJ PMEVôVOB HÌSF Y + 3y UPQMBNOOFöJUJLBÀUS \" # $ % & 6. k `3PMNBLÑ[FSF (an) = f 3n - k p EJ[JTJTBCJUEJ[JPMEVôVOBHÌSF n+4 (bn) = f kn + 20 p 4n + 5 3. F =WF' = EJ[JTJOJOEÌSEÑODÑUFSJNJLBÀUS FO = FO- + FO- O> O`; õFLMJOEF UBONMBOBO OFHBUJG PMNBZBO UBN TBZMBS \" - 4 # - $ 4 % & 7 3 3 3 EJ[JTJOEFLJTBZMBSB 'JCPOBDDJTBZMBS 'O EJ[JTJ- OFJTF'JCPOBDDJEJ[JTJEFOJS #VOBHÌSF '8 + F4 = k.F6PMEVôVOBHÌSF LLBÀ- US \" # $ % & J 49 - n2 N K n2 + 7n + 10 O 4. ( Fn 'JCPOBDDJEJ[JTJPMNBLÑ[FSF 7. _ an i = KK OO P F =WF F = L PMEVôVOBHÌSF '13LBÀUS EJ[JTJOJOLBÀUFSJNJQP[JUJGUJS A # $ % & \" # $ % & 1. A 2. A 3. C 4. A 13 5. C 6. A 7. B

TEST - 5 4BZ²SÑOUÑMFSJWF(FSÀFL4BZ%J[JMFSJ 1. #JS Bn EJ[JTJOEFrn `N+ için, 5. (an) = f 1 p n2 + 3n + 2 aO- =O ve aYBZ = PMEVôVOBHÌSF Y+ZLBÀUS EJ[JTJOJOJMLUFSJNJOJOUPQMBNLBÀUS \" # $ % & \" # 5 $ 5 2 4 % 4 & 2 5 5 2. BO = O -O+ EJ[JTJOEFBn + 1 > anFöJUMJôJOJTBôMBZBOFOLÑ- ÀÑLOUBNTBZEFôFSJLBÀUS \" # $ % & 6. a1 = 1 PMNBLÑ[FSF 2 an + 1 = an. n n 2 + JOEJSHFNFCBôOUTZMBWFSJMFO Bn EJ[JTJOJOHF- OFMUFSJNJBöBôEBLJMFSEFOIBOHJTJEJS \" 1 # 1 $ 1 n2 + n n2 n + 1 ^anh = f n2 + 4n + 18 p % 1 & n n+2 n2 + 1 n+1 3. EJ[JTJOJOUBNTBZPMBOFMemanlaSOEBOFOLÑÀÑ- ôÑOÑOEFôFSJLBÀUS \" # $ % & 7. ^ an h = f ^ - 1 hn ^ n + 2 h! p 2n 4. 2 .an = 2 + _ an i2 ve a1 = 1 CJÀJNJOEFUBONMBOBO Bn EJ[JTJJÀJO an + 1 2 an + 1 =-5 an olacaköFLJMEFQP[JUJGUFSJNMJ Bn EJ[JTJJÀJOB43 LBÀUS FöJUMJôJOJTBôMBZBOOEFôFSJLBÀUS \" # $ % 1 & 1 \" # $ % & 25 1. C 2. B 3. C 4. & 14 5. & 6. A 7. C

www.aydinyayinlari.com.tr %÷;÷-&3 4. MODÜL ·/÷7&34÷5&:&)\";*3-*, %÷;÷-&3** \"SJUNFUJL%J[J ÖRNEK 3 TANIM B C D E TPOMVBSJUNFUJLEJ[JTJOJOPSUBLGBSLEJS D+ C- B+L EBCJSTPOMVBSJUNFUJLEJ[JCFMJSUUJóJ- \"SEõLUFSJNMFSJBSBTOEBLJGBSLOTBCJUPMEVóV OFHËSF LLBÀUS EJ[JMFSFBSJUNFUJLEJ[JEFOJS b=a+2 BO EJ[JTJOEFrO` Z+J¿JO c = b + 2 = a +UÑS a - a = a - a = a - a == aO+- aO =E #VOBHÌSF D+ 3, b - 1, a + k) = (a + 7, a + 1, a +L PMVS #VEJ[JOJOPSUBLGBSL-PMEVôVOEBOB+ 1- 6 = a + k JTF BO CJSBSJUNFUJLEJ[JEJS j k = -PMVS ETBZTOBBSJUNFUJLEJ[JOJOPSUBLGBSLEFOJS #VOBHËSF ÖRNEK 4 a = a +E (FOFMUFSJNJBnPMBOCJSBSJUNFUJLEJ[JJÀJO a = a +E= a+E a =WFBO = aO- + a = a +E= a +E PMEVôVOBHÌSF EJ[JOJOHFOFMUFSJNJOJCVMVOV[ aO = aO-+E= a + O- E FõJUMJLMFSJFMEFFEJMFCJMJS a7 = a6 + 4 & a7 - a6 =E= 4 a6 = a1 +E& 24 = a1 + 20 & a1 = 4 ÖRNEK 1 a = 4 + ( n -1 ) · 4 = 4n 5BCMPEBLJJMLÑÀUFSJNJWFSJMFOBSJUNFUJLEJ[JMFSMFJMHJ- 6 MJCPöCSBLMBOZFSMFSJEPMEVSVOV[ %m/*m ÷MLÑÀUFSJN 0SUBLGBSL UFSJN a) 2, 6, 10 4 14 ²[FMMJL #JSBSJUNFUJLEJ[JEFQ! Z+WFQ<O b) 7, 5, 3 -2 1 PMNBLÐ[FSF BO = aQ + O-Q EPMVS c) 8, 8, 8 0 8 E –6, –1, 4 5 9 ²[FMMJL #JSBSJUNFUJLEJ[JOJOLUFSJNJBL e) 2, –5, –12 -7 -19 QJODJUFSJNJBQ PMNBLÐ[FSF ÖRNEK 2 ak - ap d = EJS ( an BSJUNFUJLEJ[JTJOJOUFSJNJWFPSUBLGBSLPM- EVôVOBHÌSF CVEJ[JOJO k-p a) (FOFMUFSJNJOJCVMVOV[ #VOBHËSF C UFSJNJOJCVMVOV[ aL = aQ +Ep L-Q ÖRNEK 5 UFSJNJWFUFSJNJPMBOBSJUNFUJLEJ[JOJOPS- UBLGBSLLBÀUS a) an = a1 + ( n - E a -a =d= 47 - 2 =3 22 7 = 7 + ( n - 1 ) · 3 = 3n + 4 22 - 7 15 b) a = 3· 10 + 4 = 34 10 1. a) 4, 14 b) - D E F -7, -19 15 3. -5 4. 4n 5. 3 2. a) 3n + 4 b) 34

·/÷7&34÷5&:&)\";*3-*, 4. MODÜL %÷;÷-&3 www.aydinyayinlari.com.tr ÖRNEK 6 ÖRNEK 9 BO CJSBSJUNFUJLEJ[JPMNBLÐ[FSF mJMFBSBTOBJMLUFSJNJm TPOUFSJNJPMNBL a =WFB =PMEVóVOBHËSF CVEJ[JOJO Ð[FSFUBOFEBIBUFSJNZFSMFõUJSJMFSFLCJSBSJUNFUJLEJ- a)0SUBLGBSLOCVMVOV[ [JPMVõUVSVMVZPS b)(FOFMUFSJNJOJCVMVOV[ %J[JOJOUFSJNJLBÀUS c)UFSJNJOJCVMVOV[ a1 = - 12, a14 = UFSJNZFSMFöFDFôJOEFO a) a -a =d= 13 - 5 =4 a -a 53 14 1 5-3 2 =d=2 14 - 1 b) an = a3 + ( n - 3 ) . 4 a6 = a1 + 5E= - 12 + 10 = - 2 an = 5 + 4n - 12 = 4n - 7 %m/*m c) a20 = 20 · 4 - 7 = 73 ²[FMMJL4POMVCJSBSJUNFUJLEJ[JEFCBõUBOWF ÖRNEK 7 TPOEBOFõJUV[BLMLUBCVMVOBOUFSJNMFSJOUPQMB- NCJSCJSJOFFõJUUJS ( an BSJUNFUJLEJ[JTJOEF BO = B B B BO EJ[JTJOEF a =WFB8 =EJS a + aO = a + aO- == aL + aO-L+ PMVS #VOBHÌSF EJ[JOJOLBÀODUFSJNJEJS ²[FMMJL#JSBSJUNFUJLEJ[JEFIFSUFSJNLFOEJ- TJOEFOFõJUV[BLMLUBLJUFSJNMFSJOBSJUNFUJLPSUB- an =PMTVO a -a MBNBTOBFõJUUJSL<QJ¿JO a -a 14 8 n 14 ap + k + ap – k d= = ap = 2 PMVS 6 n - 14 ÖRNEK 10 2n - 14 = 72 - 24 B C D TPOMVEJ[JTJCJSBSJUNFUJLEJ[JJTF 2n = 62 a + b +DUPQMBNLBÀUS n =PMVS 5 + 75 = a +c = 2b a + c = 80 a + b+ c = 120 ÖRNEK 8 b = 40 (an BSJUNFUJLEJ[JTJOEF a +B =+ 8a ÖRNEK 11 PMEVôVOBHÌSF B34LBÀUS ( an EPLV[UFSJNMJCJSBSJUNFUJLEJ[JPMNBLÑ[FSF a + a8 = a + 8 (a - a ) = 16 2 1 44102 4463 PMEVôVOBHÌSF B4 + a6LBÀUS 4d a2 + a8 = a4 + a6 ²[FMMJL = 16 a + 32d = 16 j a34 =CVMVOVS 1 4242 443 16 9. -2 10. 120 11. 16 a 34 6. a) 4, b) 4n - 7, c) 73 7. 31 8. 16

www.aydinyayinlari.com.tr %÷;÷-&3 4. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 12 %m/*m 4POMVCJSBSJUNFUJLEJ[JEFUFSJNJTFUFSJN #JSBSJUNFUJLEJ[JOJOJMLOUFSJNUPQMBN JMFUFSJNJOUPQMBNLBÀUS Sn = a1 + a2 + ........... + an .. a +a a1 + d a1 + (n - 1) d 9 17 = n.a1 + d (1 + 2 + . . . . . . . . . . + (n - 1)) =a 2 13 a + a = 2 · 12 = 24 d^ n - 1 hn 9 18 = n.a1 + 2 = n a 2a1 + ^ n - 1 hd k 2 ÖRNEK 13 FõJUMJóJZMFWFSJMJS ( an QP[JUJGUFSJNMJCJSBSJUNFUJLEJ[JPMNBLÑ[FSF S = n a + a + ^ n - 1 hd a + a + a + a + a + a + a = n 2 f 1 1 41 442 4 443 p a 8 n PMEVôVOBHÌSF B5LBÀUS = n ^ a + a h 2 1 n a2 + a8 = 2a5 FõJUMJóJEFFMEFFEJMJS a3 + a7 = 2a5 /PU4OCJS BO BSJUNFUJLEJ[JTJOJOJMLOUFSJN UPQMBNPMNBLÐ[FSF a4 + a6 = 2a5PMEVôVOEBO 7a = 56 j a =EJS 4O+ -4O = aO+PMVS 5 5 ÖRNEK 14 ÖRNEK 16 #JS BSJUNFUJL EJ[JOJO BSEöL ÑÀ UFSJNJ TSBTZMB #JSJODJUFSJNJWFPSUBLGBSLPMBOCJSBSJUNFUJLEJ[J- Y- Y- Y-PMEVôVOBHÌSF YLBÀUS OJOJMLPUV[UFSJNJOJOUPQMBNLBÀUS 2x - 2 + 3x - 3 30 2x - 1 = S30 = 2 ( a1 + a30 ) S30 = 15 (a1 + a1 +E 2 S30 = 15(4 + 4 + 29.3) 4x - 2 = 5x - 5 S30 = 15.(8 + 87) S30 =PMVS 3 =YPMVS ÖRNEK 15 ÖRNEK 17 Y+ +MPH Y+ ÷MLOUFSJNUPQMBN4n = n2 +OPMBOCJSBSJUNFUJL UFSJNMFSJCJSBSJUNFUJLEJ[JOJOBSEöLÑÀUFSJNJJTFY EJ[JOJOUFSJNJLBÀUS LBÀUS 2x + 3 + 4x + 1 H3 S6 - S5 = a6 = 1+ log 8 [ 22 ( 62 + 2 · 6 ) - ( 52 + 2 · 5 ) = a 6x + 4 = 8 6 2 48 - 35 = 13 x= 3 2 17 16. 1425 17. 13 12. 24 13. 8 14. 3 15. 3

·/÷7&34÷5&:&)\";*3-*, 4. MODÜL %÷;÷-&3 www.aydinyayinlari.com.tr ÖRNEK 18 ÖRNEK 21 öMLOUFSJNUPQMBN4OPMBOCJS BO BSJUNFUJLEJ[JTJJ¿JO JMFTBZMBSBSBTOEBJMFCÌMÑOFCJMFOTBZMB- 4 -å4 =WF4 -å4 =FõJUMJLMFSJWFSJMJZPS SOUPQMBNLBÀUS #VOBHÌSFEJ[JOJOHFOFMUFSJNJOJCVMVOV[ a1 = 24 a -a n1 192 - 24 an = 192 j =4 j = 4 & n = 43 n-1 n-1 S6 - S5 = a6WF410 - S9 = a10 S = 43 ·^ 24 + 192 h a –a =d= 24 – 18 3 43 2 = 10 6 10 - 6 42 S43 =PMVS 3 3n - 18 18 + 3n an = a6 + (n - 6) · 2 = 18 + 2 = 2 ÖRNEK 19 ÖRNEK 22 w EJ[JTJ JMLUFSJNJPMBOTPOMVCJS %õCÐLFZ CJS CFõHFOJO J¿ B¿MBS CJS BSJUNFUJL EJ[JOJO BS- BSJUNFUJLEJ[JEJS EõLCFõUFSJNJEJS #VEJ[JOJOUFSJNMFSJUPQMBNLBÀUS &OLÑÀÑLBÀOOÌMÀÑTÑPMEVôVOBHÌSF FOCÑ- a -a =4 j 143 - 7 =7 ZÑLBÀOOÌMÀÑTÑLBÀEFSFDFE JS n1 S5 = 540 5 n-1 n-1 540 = ( 2 · 44 +pE 143 - 7 = 4n - 4 j n = 35 2 S = 35 ^ 7 + 143 h = 2625 216 = 88 +pE 35 2 E=EJS a5 = a1 +E= 44 + 4 · 32 = 172 ÖRNEK 20 5PQMBN4FNCPMÑ %m/*m 5BUJMQMBOZBQBOCJSBJMFOJOUBUJMNBTSBGMBSBõBóEBLJUBC- G;j3 G L = aL S#OWFS O!;PMNBL MPEBWFSJMNJõUJS Ð[FSF :PM 0UFMHÐOMÐL \"MõWFSJõHÐOMÐL n 5- 5- 5- / ak = ar + ar + 1 + ar + 2 + . . . . . . . . + an olur. a) 5BUJM NBTSBG UBUJMEF HFÀFO HÑOF CBôM CJS EJ[J PMBSBLJGBEFFEJMEJôJOFHÌSF CVEJ[JOJOHFOFMUF- k=r SJNJOJCVMVOV[ #VJGBEFEFLZBJOEJTZBEBEFóJõLFO SZFalt b) 5-JMFCVBJMFFOÀPLLBÀUBNHÑOUBUJMZB- TOS OZFJTFÑTUTOSEFOJS QBCJMJS ÖRNEK 23 a) a1 = 500 + ( 250 + 50 ) · 1 a2 = 500 + ( 250 + 50 ) · 2 ++++ UPQMBNOUPQMBNTFNCPMÑJMFHÌTUFSJOJ[ an = 500 + 300 · n b) 2800 > 500 + 300 · n /10 3k 2300 > pOFOÀPLHÑOUBUJMZBQBCJMJS k =1 18 + 3n 19. 2625 20. a) 500 + 300 n b) 7 18 21. 4644 22. 172 /10 18. 23. 3k 2 k =1

www.aydinyayinlari.com.tr %÷;÷-&3 4. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 24 ÖRNEK 28 1+ 1 + 1 + ... + 1 + ... + 1 9 2 3 r 25 / ^ 7 + 3k hUPQMBNOIFTBQMBZO[ UPQMBNOUPQMBNTFNCPMÑJMFHÌTUFSJOJ[ k=1 /25 1 /9 k=1 k ^ 7 + 3k h = ^ 7 + 3 h + ^ 7 + 3.2 h + . . . + ^ 7 + 3.9 h k=1 = 10 + 13 + ... +JGBEFTJUFSJNJWFPSUBLGBSLPMBO UFSJNMJCJSBSJUNFUJLEJ[JUPQMBNES #VZÑ[EFO S = + 99 =PMVS a a a k· = (10 + 34) 9 9 2 2 1 ÖRNEK 25 3 1 ÖRNEK 29 / /5 ^ –1 hk.k m = 1 2m tPQMBNOIFTBQMBZO[ k=1 2 UPQMBNOOEFôFSJLBÀUS 11 17 ^ -1 h · 1 + 1· 2 + .... + ^ -1 h · 5 ++= 2 3 2 8 22 2 2 2 6 4471 448 6 4471 448 ^-1h+2 +^-3h+4 -5 3 = =- 22 ÖRNEK 26 ÖRNEK 30 10 ++++ / ^8–kh /UPQMBNO TFNCPMÑJMFHÌTUFSJOJ[ k=2 /13 ^ 3k + 1 h UPQMBNOOJMLJLJWFTPOJLJUFSJNJOJCVMVOV[ k =1 ÷MLJLJUFSJN L= 2 & 6 k=3& 5 4POJLJUFSJN L= 9 & -1 k = 10 & -2 ÖRNEK 27 ÖRNEK 31 /12 +++++ ^8 – kh /UPQMBNO TFNCPMÑJMFHÌTUFSJOJ[ k=1 /14 UPQMBNOIFTBQMBZO[ ^ k2 + 1 h 7 + 6 + 5 + 4 + ... + ( -3 ) + ( -4) k=3 =0 =CVMVOVS /25 1 7 19 3 /13 /14 24. 25. 26.UFS UFS UFS– UFS–2 27. 18 29. - 8 28. 198 30. ^ 3k + 1 h 31. ^ k2 + 1 h k=1 k 2 k =1 k=3

TEST - 6 \"SJUNFUJL%J[J 1. a =WFB = 5. B C D TPOMV EJ[JTJ CJS BSJUNFUJL EJ[J JTF 8 BCDÀBSQNLBÀUS \" # $ % & olan (an BSJUNFUJLEJ[JTJOJOPSUBLGBSLOFEJS \" 1 # $ 3 % & 2 2 2. )FSZM CPZVNFUSFCÐZÐZFOCJS¿OBSOCPZV 6. ÷LJODJUFSJNJ POVODVUFSJNJPMBOBSJUNF- ZMOTPOVOEBNFUSFPMNVõUVS tik EJ[JOJOCFöJODJUFSJNJLBÀUS #V ÀOBSO CPZV ZMO TPOVOEB LBÀ NFUSF \" # $ % & PMVS \" # $ % & 3. #JSBSJUNFUJLEJ[JEF 7. #JSBSJUNFUJLEJ[JEF a = B = a + a + a + a = PMEVôVOBHÌSF B18LBÀUS PMEVôVOBHÌSF JMLUFSJNUPQMBNLBÀUS \" # $ \" # $ % & % & 4. #JS Bn BSJUNFUJLEJ[JTJOEF 8. JMF BSBTOB CV TBZMBSMB CJS BSJUNFUJL EJ[J aY =Z PMVöUVSBDBLöFLJMEFUFSJNEBIBZB[MSTBCBö- aZ =Y UBOUFSJNLBÀPMVS PMEVôVOBHÌSF CVEJ[JOJOPSUBLGBSLLBÀUS \" # $ % & \" - # - $ % & 1. D 2. D 3. D 4. B 20 5. C 6. C 7. C 8. B

\"SJUNFUJL%J[J TEST - 7 1. a =WFB = 5. 5-ZFTBUMBOCJSÐSÐOJ¿JOBZ5-WFCVO- olan ( an BSJUNFUJLEJ[JTJOJOPSUBLGBSLLBÀUS EBO TPOSBLJ IFS BZ CJS ËODFLJ BZEBO 5- GB[MB PMBDBLõFLJMEFCJSËEFNFQMBOZBQMZ PS \" 1 # $ 3 % & #VOBHÌSF CVÑSÑOÑBMBOCJSLJöJFOTPOBZLBÀ 2 2 5-ÌEFZ FSFLCPSDVOVCJUJSNJöPMVS \" # $ % & 2. ¦FWSFTJVPMBOCJSÑÀHFOJOLFOBSV[VOMVLMBSCJS BSJUNFUJL EJ[JOJO BSEöL ÑÀ UFSJNJ JTF PSUBODB LFO BS OV[VOMVôVBöBôEBLJMFSEFOIBOHJTJEJS \" u # u $ 3u % 2u & V 6. #JS Bn BSJUNFUJLEJ[JTJOEF 3 2 23 a =Y B =Z PMEVôVOBHÌSF B30LBÀUS \" Z-Y # Z-Y $ Y+Z % Y+Z & y - x 2 3. #JMHJBY +CY +DY+E=EFOLMFNJOJOLËLMFSJ Y Y YPMNBLÐ[FSF x1 + x2 + x3 =- b ES a Y +Y +NY+N+= 7. #JS ( an EJ[JTJOJOJMLOUFSJNUPQMBN EFOLMFNJOJOLÌLMFSJCJSBSJUNFUJLEJ[JPMVöUVSEV- Sn = n2 + 3 n ôVOBHÌSF NLBÀUS 2 PMEVôVOBHÌSF B5 + a6UPQMBNLBÀUS \" # $ % & \" # $ % & 4. JMFTBZMBSBSBTOBCVTBZMBSMBCJSMJLUFBSJU- 8. ( a CJSBSJUNFUJLEJ[JPMNBLÑ[FSF NFUJL EJ[J PMVöUVSBDBL öFLJMEF UFSJN EBIB n ZFSMFöUJSJMJSTFPMVöBOEJ[JOJOUFSJNJBöBôEBLJ- MFSEFOIBOHJTJEJS a - a =WFB + a = \" # $ % & oMEVôVOBHÌSF B4LBÀUS \" # $ % & 1. D 2. A 3. A 4. A 21 5. D 6. A 7. A 8. &

TEST - 8 \"SJUNFUJL%J[J 1. #JS BO BSJUNFUJLEJ[JTJOJOJMLOUFSJNJOJOUPQMBN4O 5. #JS BEBNO TFLJ[ HÐOMÐL IBSDBNBMBS CJS BSJUNFUJL EJS EJ[JOJOBSEõLTFLJ[UFSJNJEJSöMLHÐOLÐIBSDBNBT 5-EJS a = 4 -4 = 5PQMBNIBSDBNBT5-PMEVôVOBHÌSF TPO PMEVôVOBHÌSF PSUBLGBSLLBÀUS HÑOLÑIBSDBNBTBöBôEBLJMFSEFOIBOHJTJEJS \" # 3 $ % 4 & 3 \" # $ % & 2 54 2. #JSBSJUNFUJLEJ[JEF 6. i:MOFOJZJEJ[JTJuËEÐMÐOÐBMNBZBTBIOFZF¿LBO a + a + a+ a = EJ[J FLJCJOJO LJõJMJL LBESPTVOVO PZVODVMBSOO PMEVôVOBHÌSF JMLUFSJNUPQMBNLBÀUS ZBõMBS CJS BSJUNFUJL EJ[J PMVõUVSNBLUBES #V PZVO- DVMBSO ZBõMBS UPQMBN PMVQ FO ZBõMMBS PMBO \" # $ PZVODVZBõOEBES % & #VOBHÌSF FOLÑÀÑLPZVODVLBÀZBöOEBES \" # $ % & 3. BO BSJUNFUJLEJ[JTJOJOBSEõLUFSJNJY Y+Z 7. ( a CJSBSJUNFUJLEJ[JPMNBLÑ[FSF Y+[EJS CO BSJUNFUJLEJ[JTJOJOJMLUFSJNJZ+[ n PMVQPSUBLGBSLZ-[EJS #VOBHÌSF Cn EJ[JTJOJOJMLUFSJNJOJOUPQMB- a + a + a ++ a =WF NBöBôEBLJMFSEFOIBOHJTJEJS a + a + a ++ a = \" [ # [ $ [ PMEVôVOBHÌSF B50- a25GBSLLBÀUS % [ & [ \" # $ % & 4. ( an BSJUNFUJLEJ[JTJJÀJO 14 24 8. \"MUODUFSJNJWFPOEÌSEÑODÑUFSJNJPMBO / /ak = 88 ve ak = 360 CJSBSJUNFUJLEJ[JEFJMLUFSJNJOUPQMBNLBÀUS k=4 k = 10 \" # $ % & PMEVôVOBHÌSF ( aO EJ[JTJOJOUFSJNJLBÀUS \" # $ % & 1. A 2. C 3. & 4. D 22 5. B 6. C 7. D 8. &

5PQMBN4FNCPMÑ TEST - 9 1. ++++ 4. #JMHJO! N+WFC`3PMNBLÐ[FSF UPQMBNOO ! TFNCPMÑ JMF HÌTUFSJNJ BöBôEBLJ- / /n n MFSEFOIBOHJTJEJS _ b·k i = b k ES k =1 k =1 12 20 23 23 /\" 4k + 1 /# ^ 4k + 9 h / /b · ak = 48 ve ap = 12 k =1 k =1 k =1 p=1 20 21 PMEVôVOBHÌSF CLBÀUS /$ 4k + 9 /% _ 4k + 5 i \" # $ % & k =1 k =1 21 /& _ 4k + 1 i k =1 5. Y -Y+=EFOLMFNJOJOLËLMFSJYWFYPM- NBLÐ[FSF 2 1 / k =1 xk 2. ++++ UPQMBNOOEFôFSJLBÀUS UPQMBNOO ! TFNCPMÑ JMF HÌTUFSJNJ BöBôEBLJ- \" 5 # $ 4 % 3 & - 4 MFSEFOIBOHJTJPMBNB[ 4 5 55 /14 /12 \" ^ 5k - 3 h # _ 5k + 7 i k =1 k =-1 14 16 /$ _ 5k + 2 i /% ^ 5k - 13 h k=0 k=3 6. 4FLJ[JODJUFSJNJ POZFEJODJUFSJNJPMBOCJS /10 BSJUNFUJL EJ[JOJO HFOFM UFSJNJ BöBôEBLJMFSEFO IBOHJTJEJS & _ 5k + 17 i k = –3 \" O+ # O+ $ O- % O- & O+ /24 7. ÷MLOUFSJNUPQMBN4n = 2n2 +OPMBOCJSBSJUNF- 3. ^ –1 hk^ 2k - 3 h UJLEJ[JOJOHFOFMUFSJNJBöBôEBLJMFSEFOIBOHJTJ- k =1 EJS UPQMBNOOEFôFSJLBÀUS \" # $ \" O+ # O+ $ O+ & - % - % O+ & O- 23 1. B 2. C 3. B 4. C 5. C 6. B 7. C

·/÷7&34÷5&:&)\";*3-*, 4. MODÜL %÷;÷-&3 www.aydinyayinlari.com.tr %÷;÷-&3*** (FPNFUSJL%J[J %m/*m TANIM ²[FMMJL BO HFPNFUSJLEJ[JTJOEFGL<O WFL! Z+PMNBLÐ[FSFPSUBL¿BSQBOSJTF \"SEõLUFSJNMFSJBSBTOEBLJPSBOTBCJUPMBOEJ- aO = aLpSO-LPMVS [JMFSFHFPNFUSJLEJ[JEFOJS r ` R - { } PMNBLÐ[FSFrO` Z+J¿JO aaa a 2 = 3 = 4 = ....= n+1 = r aaa a 123 n ÖRNEK 3 JTF BO CJSHFPNFUSJLEJ[JEJSSHFS¿FLTBZTCV HFPNFUSJLEJ[JOJOPSUBL¿BSQBOESöMLUFSJNJB r ` R+WF BO CJSHFPNFUSJLEJ[JPMNBLÐ[FSF B = WFPSUBL¿BSQBOSPMBO BO HFPNFUSJLEJ[JTJOJO ve a =PMEVóVOBHËSF CVEJ[JOJO HFOFMUFSJNJ a ) 0SUBLÀBSQBOOCVMVOV[ b ) (FOFMUFSJNJOJCVMVOV[ aO = apSO-PMVS c ) UFSJNJOJCVMVOV[ ÖRNEK 1 a) a4 = a2pS2 8 =pS2 & S2 = 2 j S = 2 ÷MLÑÀUFSJNJWFSJMFOHFPNFUSJLEJ[JMFSJMFJMHJMJCPöC- SBLMBOZFSMFSJEPMEVSVOV[ n+2 ÷MLÑÀUFSJN 0SUBLÀBSQBO UFSJN b) an = a2pSn-2 = 4·^ 2 hn–2 = 2 2 1 5 2 4 n+2 12 2 56 a) 5 16 45 c) a = 2 2 = 2 2 = 64 9 10, 5, fp 10 1 3 2 5 b) 7, 14, 28 c) 3 4 64 ,, 4 3 27 E 5, 5, 5 ÖRNEK 2 ÖRNEK 4 BO HFPNFUSJLEJ[JTJOJOUFSJNJWFPSUBL¿BSQBO 1 #JS HFPNFUSJL EJ[JOJO BSEöL ÑÀ UFSJNJ TSBTZMB 2 Y- Y Y+PMEVôVOBHÌSF YLBÀUS PMEVóVOBHËSF CVEJ[JOJO a ) (FOFMUFSJNJOJCVMVOV[ aa 5x 5x + 20 b ) UFSJNJOJCVMVOV[ 23 =j = aa 5x - 4 5x a) an = a1pSn-1 12 1 n-1 4 - n 25x2 = ( 5x - 4 ) ( 5x + 20 ) 2 25x2 = 25x2 + 80x - 80 an = 8 ·f p = 2 a6 = 24-6 = 2-2 = 1 80 = 80x j 1 =YPMVS 4 b) 15 b) 2, 56 c) 16 ,d 4 5 E 2. a) 24–n 1 24 n+2 1. a) , b) 93 n 4 3. B S 2 b) an= 2 2 c) 64 4. 1 24

www.aydinyayinlari.com.tr %÷;÷-&3 4. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 5 %m/*m 1P[JUJG UFSJNMJ Bn EJ[JTJ a5 = 1 WF B3 + a6 = 45 ²[FMMJL4POMVCJSHFPNFUSJLEJ[JEFCBõUBOWF a7 9 TPOEBOFõJUV[BLMLUBCVMVOBOUFSJNMFSJO¿BSQ- NCJSCJSJOFFõJUUJS FöJUMJLMFSJOJTBôMBZBOCJSHFPNFUSJLEJ[JJse a1LBÀUS apBO = apBO- = apBO- == aLpBO-L+ #VOBHËSF O<LPMNBLÐ[FSF a 1 an2 = aO-LpBO+LPMVS = &r=3 5 ÖRNEK 8 2 9 B C D TPOMVEJ[JTJCJSHFPNFUSJLEJ[JJTFBpCpD ÀBSQNLBÀUS a ·r 5 a3 + a6 = a1p S2 +S5 ) = 45 a1 · (9 + 243 ) = 45 a1 = 45 5 = 252 28 ÖRNEK 6 7 · 63 = a · c = b2WFBpCpD= b3UÑS#VOBHÌSF b2 = 72 · 32 & b =WFBpCpD= 213PMBSBLCVMVOVS NV[VOMVóVOEBCJSJQIFSCJSJCJSHFPNFUSJLEJ[JOJOBS- EõLUFSJNMFSJPMBDBLõFLJMEFQBS¿BZBCËMÐOÐZPS ÖRNEK 9 &OLÑÀÑLQBSÀBFOCÑZÑLQBSÀBOO 1 JPMEVôVOB BY +CY +DY+E=EFOLMFNJOJOLËLMFSJY YWFY 8 HÌSF FOLÑÀÑLQBSÀBOOV[VOMVôVLBÀNEJS a4 = a1pS3 & a1pS3 · 1 = a1 &S= 2 PMNBLÐ[FSF YpYpY = - d ES 8 a #VOBHÌSF Y +Y -BY+ 8 = x + 2x +4x + 8x = 45 EFOLMFNJOJOLÌLMFSJCJSHFPNFUSJLEJ[JOJOBSEöLÑÀ & 15x = 45 x=3 UFSJNJJTFBLBÀUS ÖRNEK 7 x1 · x2 · x3 = -WFY1 · x3 = x22PMEVôVOEBO 5- BZML CJMFõJL GBJ[ WFSFO CJS CBOLBZB ZBUS- x23 = -8 j x2 = -EJSY2 = -EFOLMFNJOJOLÌLÑPMEVôVO- MZPS EBOZFSJOFZB[EôN[EBEFOLMFNJTBôMBS#VEVSVNEB OBZTPOSBCVQBSBGBJ[JZMFCJSMJLUFLBÀ5-PMBSBLÀF- -8 + 8 + 2a + 8 = 0 & a = -PMVS LJMJS ÖRNEK 10 101 a1 = 2000 · 100 ÷MLUFSJNJOJOÀBSQNPMBO Bn HFPNFUSJLEJ[JTJ için a2 · a5LBÀUS 101 101 a2 = = 2000 · 100 G · 100 a1 · a2 · a3 · a4 · a5 · a6 = 64 j a .a = a .a = a .a 101 n 16 25 34 an = 2000 · d 100 n PMVS PMEVôVOEBO B2 · a5 )3 = 64 & a2 · a5 =PMVS 6. 3 7. 2000 · d 101 n 25 8. 213 9. –4 10. 4 5. 100 n 28

·/÷7&34÷5&:&)\";*3-*, 4. MODÜL %÷;÷-&3 www.aydinyayinlari.com.tr ÖRNEK 11 ÖRNEK 14 ( an CJSHFPNFUSJLEJ[JWFaBBBB8 = BO CJSHFPNFUSJLEJ[JPMNBLÐ[FSF B =WFa = 1 PMEVôVOBHÌSF B6LBÀUS 9 8 a4 · a8 = a5 · a7 = a62 a62 · a6 · a62= 243 & ( a6 )5 = 243 & a6 = 3 PMEVóVOBHËSF CVEJ[JOJO a) 0SUBLÀBSQBOOCVMVOV[ b) (FOFMUFSJNJOJCVMVOV[ c) UFSJNJOJCVMVOV[ ÖRNEK 12 a) r = 8 – 3 a = 1 8 a 3 3 BO HFPNFUSJLEJ[JTJOEF B + a += a ve b) an = a8pSn-8 = 36-n aBB =EJS c) a20 = 3-14 0SUBLÀBSQBOQP[JUJGJTFB3LBÀUS ( a2 )2 a1 · a2 · a3 = ( a2 )3 = 8 & a2 =EJS a + 2 + = =S B3 = a2pSWFB1, = 2 a 1 a r) 1 3 2 ÖRNEK 15 r + 3 =S B1pS= 2 = a2 ) 2 +S=S2 &S= 2 a3 = 2 · 2 =UÑS 8 UÐS 3 BO HFPNFUSJLEJ[JTJOEFB = B = %m/*m #VEJ[JOJOPSUBLÀBSQBOOFEJS ²[FMMJLL>Q QP[JUJGUFSJNMJCJSHFPNFUSJLEJ- r = 7–4 a 2 [JEFLUFSJNBL QUFSJNBQPMNBLÐ[FSFEJ[JOJO 7 PSUBL¿BSQBO a = 3 r = k–p ak PMVS 4 ap ÖRNEK 13 %m/*m #FöJODJUFSJNJ 1 TFLJ[JODJUFSJNJ 1 PMBOHFPNFU- 2 16 SJLEJ[JOJOJLJODJUFSJNJLBÀUS a -4 öMLUFSJNJBWFPSUBL¿BSQBOSPMBO BO HFP- 8 =3 21 NFUSJLEJ[JTJOJOJMLOUFSJNJOJOUPQMBN r = 8-5 = a -1 2 4O = a + aS++ aSO- 5 2 a 1 ^ 1 - rn h 1-r 1 = 5-2 5 & 1 = 2 & a = 4 tür. Sn = a1· EJS 2 a 8 a 2 2 2 11. 3 12. 4 13. 4 26 1 b) a = 36 – n c) a = 3–14 2 14. a) r = 15. n 20 3 3

www.aydinyayinlari.com.tr %÷;÷-&3 4. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 16 ÖRNEK 19 ÷MLUFSJNJ PSUBLÀBSQBO 1 PMBOCJSHFPNFUSJLEJ[J- ^anh = f 2 p 3 3n + 2 TJOJOJMLUFSJNJOJOUPQMBNOFEJS HFPNFUSJLEJ[JTJOJOJMLOUFSJNUPQMBNLBÀUS 1– d 1 8 3 n S =4· 2 81 1– 3 a 43 2 3 23 1 =r= =·= 88 a 2 4 2 3 =4· 3 ·f 3 –1 p=2·f 3 –1 p 1 3 2 8 7 3 3 3 3 1– d 1 n n n 2 3 = 2 · 3 ·f 3 –1 p S= · n n3 1 3 2 1– 3 3 3 3 n = 1 f 3 – 1 p olur. ÖRNEK 17 2 n BO HFPNFUSJLEJ[JTJOEF B =YWFBO =ZEJS 3 3 #VEJ[JOJOPSUBLÀBSQBOPMEVôVOBHÌSF JMLOUFSJN UPQMBNOOYWFZDJOTJOEFOJGBEFTJOFEJS ÖRNEK 20 an = a1pSn-1 & y = x · 2n-1 = 2y = 2nPMVS ( an TPOMVHFPNFUSJLEJ[JTJOEFJMLUFSJNPSUBLÀBS- x QBOWFJMLUFSJNUPQMBNJTFCVEJ[JOJOUF- ^ 1– rn h S =a · SJNTBZTLBÀUS n 1 1– r ^ 1 – n h =x·d 2y –1n ^ 1 – n h x =x· 2 r Sn = 3640 = 10 · 1– r ^ –1 h = 2y -YPMVS n j n =ES j 364 = 3 – 1 j 3n = 729 3–1 ÖRNEK 18 ÖRNEK 21 #JSHFPNFUSJLEJ[JEF JMLOUFSJNUPQMBN #JS HFPNFUSJL EJ[JEF JML UFSJN UPQMBNOO JML Ð¿ UFSJN 4O =O+ - UPQMBNOBPSBOEJS PMEVôVOBHÌSF CVEJ[JOJOUFSJNJLBÀUS #VOBHÌSF EJ[JOJOPSUBLÀBSQBOOFEJS 1– r 6 S a· ^ 1– r 3 h^ 1+ 3 h 1 1– r 6 = = 28 & r = 28 S10 - S9 = a10 S 3 1– r 3 & ( 212 - 4 ) - ( 211 - 4 ) = 212 - 211 3 1– r = 211 a· 1 1– r & 1 +S3 = 28 &S= 3 8 n 27 19. 1 f 3 - 1 p 16. 2·f 3 –1 p 17. 2y – x 18. 211 20. 6 21. 3 7 n 3 2 3 3

·/÷7&34÷5&:&)\";*3-*, 4. MODÜL %÷;÷-&3 ÖRNEK 24 www.aydinyayinlari.com.tr %J[JMFSMF÷MHJMJ(FSÀFL)BZBU1SPCMFNMFSJ 1BST LÐQõFLMJOEFLJPZVODBL LVUVMBSOIFSTSBEBCJSBMUT- ÖRNEK 22 SBEBLJOEFOCJSFLTJLLVUVPMB- DBL õFLJMEF ZBOEBLJ HJCJ EJ[J- #JSCBLUFSJ¿FõJEJOJOQPQÐMBTZPOVVZHVOPSUBNEBIFS ZPS EBLJLBEBCJSLBULBEBSBSUNBLUBES #BöMBOHÀUB CBLUFSJOJO CVMVOEVôV CV PSUBNEB TBBUJOTPOVOEBUPQMBNLBÀCBLUFSJCVMVOVS 20 saat =pEBLJLB a) &OBMUTSBEBLÑQPMEVôVOEBUPQMBNLBÀLÑQ 20 · 60 LVMMBONöPMVS = 30 UFSJNPMVöBDBLUS b) 5PQMBNLÑQLVMMBOMEôOEBLBÀTSBEJ[JMNJö 40 PMVS a1 =p LBUBSUBDBôOEBO a2 = 120.3.3 = 120.32 c) 1BSTUPQMBNLÑQLVMMBOEôOEB J ,BÀLÑQLVMMBOMBNB[ a30 = 120 · 331PMVS JJ 5PQMBNLBÀTSBEJ[NJöPMVS %m/*m B TSB\" 1 18 TSB\" 2 S18 = 2 ( 1 + 18 ) i SBEZBO PMNBL Ð[FSF iiu SBEZBOML NFSLF[ B¿OO PMVõUVSEVóV iSu ZBS¿BQM EBJSF EJMJNJOJO TSB\" 18 = 9 · 19 = 171 BMBO 1 r iJMFCVMVOVS b) S= n a 1+ a k = 465 2 n 2 n & n · ^ n + 1 h = 465 & n^ n + 1 h = 930 & n = 30 2 ÖRNEK 23 D OTSBPMVöUVSNBLÑ[FSF n^ n + 1 h # 600 & n = 34 %BJSFTFMCJSEJTL BMBOMBSBSJUNFUJLEJ[JPMVõUVSBDBLõFLJM- EFEJMJNFBZSMZPS 2 &O CÑZÑL EJMJNJO NFSLF[ BÀTOO ÌMÀÑTÑ FO LÑÀÑL EJMJNJO NFSLF[ BÀTOO ÌMÀÑTÑOÑO LBU PMEVôVOB TSBZMBUPQMBNLÑQLVMMBOMS LÑQBSUBS HÌSF FO LÑÀÑL EJMJNJO NFSLF[ BÀTOO ÌMÀÑTÑ LBÀ SBEZBOES ÖRNEK 25 12 2 LNMJLCJSNBSBUPOLPõVTVJ¿JOBOUSFONBOZBQBOLP- S = (a + a ) = rr õVDV CJSJODJ HÐO LN LPõBSBL BOUSFONBOB CBõMZPS ,PõNBNFTBGFTJOJIFSHÐO 1 PSBOOEBBSUUSBSBLBO- 12 2 1 12 EJTLJONFSLF[BÀTOOÌMÀÑTÑiSBEZBOPMNBLÑ[FSF 2 USFNBOOBEFWBNFEJZPS a= 2 , a= ^ 2i hr2 ,BÀ HÑOÑn TPOVOEB QBSLVSVO UBNBNO LPöNVö ir PMVS f log 3 10 . 5, 67 p 12 12 2 2 2 ir PMEVôVOEBOB12 = 2a1 (i SBEZBOMLEJMJNJOBMBO 2 a1 = 2 3 EJS , a2 = 2· 2 = 3 = 6 · f ir 22 2ir 32 3 n–1 #VOBHÌSF S + p = rr2 a = 2·d n , a = 2·d n PMNBLÑ[FSF 12 2 2 32 n2 ir2 = rr2 & i = r 3 n–1 9 2·d n = 20 j n - 1 = log 3 10 jOáPMVQ 2 2 HÑOTPOVOEBQBSLVSVOUBNBNOLPöNVöPMVS 22. 120 · 331 π 28 24. a) 171 b) 30 c) i. 5, ii. 34 25. 7 23. 9

www.aydinyayinlari.com.tr %÷;÷-&3 4. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 26 ÖRNEK 28 #JS õJSLFU ¿BMõBOMBSOB JLJ GBSLM NBBõ ËEFNF TF¿FOFóJ õFLJM õFLJM õFLJM TVOVZPS ZMML \" WF # TF¿FOFLMFSJZMF JMHJMJ BõBóEBLJ CJMHJMFSWFSJMNFLUFEJS \"4FÀFOFôJ r #JSJODJZM5-WFIFSZM5-MJLBSUõ #4FÀFOFôJ r #JSJODJZM5-WFIFSZMMJLBSUõ #VCJMHJMFSFHÌSF õFLJM a) \"TFÀFOFôJOJUFSDJIFEFOLJöJOJOZMTPOVOEB- :VLBSEBLJ ÌSÑOUÑZF HÌSF BöBôEBLJ TPSVMBS ZBOU- LJNBBöOFPMVS MBZO[ a) öFLJMEFLJLBSFTBZTOFPMBDBLUS b) #TFÀFOFôJOJUFSDJIFEFOLJöJOJOZMTPOVOEB- b) #VÌSÑOUÑOÑOHFOFMUFSJNJOJCVMVOV[ LJNBBöOFPMVS à BMO[ B öFLJM\" 1 a) a = E= 600 öFLJM\" 1 + 3 = 1 + 3 · 1 = 4 öFLJM\" 1 + 3 + 3 = 1 + 3 · 2 = 7 1 öFLJM\" 1 + 3 + 3 + 3 = 1 + 3 · 3 = 10 öFLJM\" 1 + 3 ·4 =PMVS a10 = 5000 + 9 · 600 b) a1 = E= 3 = 10400 an = 1 + (n - 1) ·3 = 3n -PMVS b) a1 = 5000 a2 = 5000 · 1,08 a = (5000 · 1,08) · 1,08 = 5000 · (1,08)2 3 a10 = 5000 · (1,08)9 = 9950 ÖRNEK 27 ÖRNEK 29 #BUVCJSLºóEBËODFZB[ZPS TPOSBBMUTSBZBHF¿FSFL &SFOIFTBCOEBLJQBSBOOÐ¿HÐOEFCJS 1 TJOJ¿FLJZPS WF TPOSB CJS BMU TSBZB HF¿FSFL ZB[ZPS #V 2 ËSÐOUÐZÐOTBUSBOUBOFTBZZB[BSBLEFWBNFUUJSJZPS TSBEBLJTBZMBSOUPQMBNLBÀUS HÑO TPOVOEB IFTBCOEB 5- LBMEôOB HÌSF CBöMBOHÀUBOFLBEBSQBSBTWBSES 1 ZTSBOOTPOTBZT 3 ZTSBOOTPOTBZT+ 2 #BöMBOHÀUBLJQBSBYPMTVO 6 ZTSBOOTPOTBZT+ 2 + 3 1 a1 = x · 2 HÑOTPOV 10 ZTSBZ 1 + 2 + 3 + 4 j 10 - 4 + 1 = 7 11 #VOB HÌSF TSBOO TPO TBZT 18.19 = 171 PMVS a = x · 2 · 2 HÑOTPOV 2 2 TSBOOTPOTBZT 17.18 = 153 UÑS 1 a5 = x · 32 = HÑOTPOV 2 x =CVMVOVS #VTSBEBLJJMLTBZTPOTBZPMVQ 154 + 155 + ... + 171 =PMVS 26. a) 10400, b) 9950 27. 2925 29 28. a) 13, b) 3n – 2 29. 6400

TEST - 10 (FPNFUSJL%J[J 1. #JS Bn HFPNFUSJLEJ[JTJOEF 5. #JS HFPNFUSJL EJ[JEF JML BMU UFSJNJO UPQMBN JML a =WFB = ÑÀ UFSJNJO UPQMBNOO LBUOB FöJU JTF EJ[JOJO PMEVôVOBHÌSF B20OFEJS PSUBLÀBSQBOBöBôEBLJMFSEFOIBOHJTJEJS \" # 3 2 $ 3 4 % & 2 3 2 \" # $ % & 2. #JSHFPNFUSJLEJ[JEFB9 = 64 . a3CBôOUT 6. #JSHFPNFUSJLEJ[JOJOBSEöLÑÀUFSJNJTSBTZMB WBSTBPSUBLÀBSQBOOFEJS a - B+ B+ PMEVôVOBHÌSF BLBÀUS \" 1 # 1 $ 2 % 3 & 3 232 \" # $ % & 3. #JMHJBY +CY +DY+E=EFOLMFNJOJOLËLMFSJ 7. 1P[JUJG UFSJNMJ CJS HFPNFUSJL EJ[JOJO UFSJNJ Y YWFYPMNBLÐ[FSF YYY = - d ES 1 WFUFSJNJPMEVôVOBHÌSF CVEJ[JOJO a 27 Y -Y +BY+= UFSJNJLBÀUS EFOLMFNJOJOLÌLMFSJCJSHFPNFUSJLEJ[JOJOJMLÑÀ \" 1 # $ 4 % & UFSJNJJTFBLBÀUS 3 3 \" - # - $ % & 8. #JSHFPNFUSJLEJ[JEF 4. BO QP[JUJGUFSJNMJCJSHFPNFUSJLEJ[JPMNBLÐ[FSF a =WFB = PMEVôVOBHÌSF B20OFEJS EJ[JOJOJMLUFSJNJOJO¿BSQNUFSJNJOJOLBUES #VOBHÌSF B5LBÀUS \" # $ % & \" 4 2 # $ % & 1. & 2. C 3. A 4. B 30 5. & 6. B 7. D 8. D

(FPNFUSJL%J[J TEST - 11 1. #JSCBOLBZB5-QBSB ZMMLCJMFõJLGBJ[- 5. 1P[JUJGUFSJNMJBSUBOCJSHFPNFUSJLEJ[JOJOBSEöL MFZMMóOBZBUSMZPS EÌSU UFSJNJOEFO JML JLJTJOJO GBSL TPO JLJTJOJO GBSLPMEVôVOBHÌSF CVEJ[JOJOPSUBLÀBSQBO #VOB HÌSF IFS ZMO TPOVOEB CBOLBOO IFTB- LBÀUS COEBHÌSÑMFOCBLJZFMFSJOUPQMBNLBÀ5-PMVS \" # $ \" # $ % & & % 6. ¥FWSFTJCJSJNPMBOCJSLBSFOJOJ¿JOF¿FWSFTJEõUB- LJLBSFOJO¿FWSFTJOJOZBSV[VOMVóVOEBPMBOJ¿J¿F LBSFEBIB¿J[JMJZPS 2. #JSFMFNFOUJOZBSMBONBTÐSFTJHÐOEÐS &MEFFEJMFOUPQMBNLBSFOJOÀFWSFV[VOMVLMBS UPQMBNLBÀCJSJNEJS #VFMFNFOUJOHÑOTPOVOEBLBÀUBLBÀZBS- MBOBSBLLBZCPMNVöUVS \" 1- 1 # 1- 1 $ 1- 1 \" 29 # 30 $ 31 % 32 & 33 29 210 211 30 31 32 33 34 % 2 - 1 & 2 – 1 29 210 3. ( an QP[JUJGUFSJNMJCJSHFPNFUSJLEJ[JPMNBLÑ[FSF 7. x, y !3+PMNBLÑ[FSF a = BO = YZ YZ YZ a7 + a5 = 5 EJ[JTJOJO IFN BSJUNFUJL IFN EF HFPNFUSJL EJ[J a7 - a5 PMNBTJÀJO Y+ZLBÀPMNBMES PMEVôVOBHÌSF B9 kaÀUS \" # $ % & \" 3 # 9 $ 51 % 27 & 81 2 4 9 24 8. #JSHFPNFUSJLEJ[JOJOJMLUFSJNJ PSUBL¿BSQBOOBFõJU- UJS 4. #JSHFPNFUSJLEJ[JOJOÐ¿ÐODÐUFSJNJBMUODUFSJNJOJO #VEJ[JOJOJMLPOUFSJNJOJOUPQMBN 210 – 1 LBUES 210 UFSJNJ 1 PMBOCVEJ[JOJOJMLUFSJNJLBÀUS 28 PMEVôVOBHÌSF PSUBLÀBSQBOLBÀUS A # $ % & \" 1 # 1 $ 1 % 1 & 1 9 8 4 32 1. C 2. C 3. & 4. B 31 5. A 6. D 7. C 8. &

TEST - 12 (FPNFUSJL%J[J 1. #JSHFPNFUSJLEJ[JOJOBSEöLÑÀUFSJNJB- 6, a - 3, 5. B JMF C TBZMBS BSBTOB CV TBZMBSMB CJSMJLUF CJS a +PMEVôVOBHÌSF EJ[JOJOCJSTPOSBLJUFSJNJ HFPNFUSJLEJ[JPMVöUVSVMBDBLöFLJMEFUFSJNEB- BöBôEBLJMFSEFOIBOHJTJEJS IBZFSMFöUJSJMJSTFCVEJ[JOJOPSUBLÀBSQBOBöBô- EBLJMFSEFOIBOHJTJPMVS $ 27 % 81 & \" # 22 \" 7 b # 8 b $ 7 a + b a a b % c a 7 & f b 8 b m a p 2. ·ÀÑODÑUFSJNJ JMLUFSJNJOEFOGB[MB EÌSEÑO- 6. TJO B TJO CJS HFPNFUSJL EJ[JOJO BSEöL DÑ UFSJNJ JLJODJ UFSJNJOEFO GB[MB PMBO HFP- UFSJNJPMEVôVOBHÌSF BBöBôEBLJMFSEFOIBO- NFUSJLEJ[JOJOCFöJODJUFSJNJLBÀUS HJTJPMBCJMJS \" # $ & % \" TJO # TJO $ TJO % TJO & TJO 3. #JSLBQUBCVMVOBOLÐMUÐSCBLUFSJTJIFSHÐOÐOTPOVO- EBLBUOB¿LBSBL¿PóBMZPS #JSJODJHÑOÑOTPOVOEBLBQUBUBOFLÑMUÑSCBL- n2 m 3.n UFSJTJPMEVôVOBHÌSF HÑOÑOTPOVOEBLBQUB LBÀUBOFLÑMUÑSCBLUFSJTJPMNVöPMVS 7. ÷MLOUFSJNJOJOÀBSQN¦n = 3 2 PMBOCJS HFPNFUSJLEJ[JEFCFöJODJUFSJNLBÀUS \" + # + $ - \" # $ % & & % - 4. 1P[JUJGUFSJNMJCJSHFPNFUSJLEJ[JEFJMLUFSJNJO 8. #JS HFPNFUSJL EJ[JEF EPLV[VODV UFSJNJO EÌS- UPQMBNOO JML UFSJN UPQMBNOB PSBO PMEV- EÑODÑUFSJNFPSBOPMEVôVOBHÌSF TFLJ[JODJ ôVOB HÌSF CV EJ[JOJO PSUBL ÀBSQBO BöBôEBLJ- UFSJNJOBMUODUFSJNFPSBOLBÀUS MFSEFOIBOHJTJEJS \" # $ 2. 3 2 & 4 2 \" 4 17 # 17 $ % % 2 17 & 1. D 2. B 3. C 4. C 32 5. B 6. B 7. C 8. D

(FPNFUSJL%J[J TEST - 13 1. \"öBôEBLJMFSEFO IBOHJTJ CJS HFPNFUSJL EJ[JOJO 5. öML ZÐLTFLMJóJ DN PMBO CV[ LÐUMFTJ IFS EBLJLB JMLOUFSJNUPQMBNPMBCJMJS 1 ÐLBEBSFSJNFLUFEJS 3 \" 4O=O # 4O =O #VCV[LÑUMFTJOJOCPZVIBOHJEBLJLBMBSBSBTO- $ 4O = O+ % 4O =O+ EBDNOJOBMUOBJOFS & 4O = O- \" - # - $ - % - & - 2. _ an i = f 1 + 1 + 1 +...+ 1 p 6. #JSLBQMVNCBóBJMLTBBUUFNZPMHJEJZPS#VOEBO 5 52 53 5n TPOSBLJIFSTBBUUFCJSËODFBMEóZPMVOZBSTLBEBS EJ[JTJOEF BMUOD UFSJNJO ÑÀÑODÑ UFSJNF PSBO ZPMHJEJZPS LBÀUS #VOBHÌSF TBBUJOTPOVOEBUPQMBNLBÀNFUSF ZPMHJUNJöUJS \" 128 # 126 $ 118 \" 75 # 115 $ 245 125 125 125 8 8 8 % 116 % 635 125 & 112 8 & 715 125 8 3. (an QP[JUJGUFSJNMJCJSHFPNFUSJLEJ[JPMNBLÑ[F- 7. ( an HFPNFUJLEJ[JTJJÀJO SF 2 . a5 . a17 + a18 . a4 + a10 . a12 3a7 . a15 a - a =B PSBOLBÀUS PMEVôVOBHÌSF EJ[JOJOPSUBLÀBSQBOLBÀUS \" # $ % & \" 1 # 2 $ % 4 & 5 3 3 3 3 4. 1P[JUJGUFSJNMJCJS Bn HFPNFUSJLEJ[JTJOEF 8. WF TBZMBSOO IFS CJSJOF BZO TBZ FLMFO- aB = EJóJOEF HFPNFUSJL EJ[JOJO BSEõL JML Ð¿ UFSJNJ FMEF aB = FEJMJZPS PMEVôVOBHÌSF B5LBÀUS #VOBHÌSF HFPNFUSJLEJ[JOJOUFSJNJBöBôEB- LJMFSEFOIBOHJTJEJS \" # $ % & \" 2 5 # 4 5 $ & % 1. & 2. B 3. C 4. B 33 5. A 6. D 7. D 8. D

TEST - 14 (FSÀFL)BZBU1SPCMFNMFSJ WF TPSVMBS BøBôEBLJ CJMHJMFSF HÌSF WF TPSVMBS BøBôEBLJ CJMHJMFSF HÌSF DFWBQMBZO[ DFWBQMBZO[ ZMOEB\"BEBTOEBZBõBZBOYUÐSÐOFBJUDBO- #JSLPMUVLÐSFUJNGBCSJLBTOEBBZLPMUVLÐSFUJ- MMBSO TBZT EJS UFO JUJCBSFO Y UÐSÐOF MJZPS#VOEBOTPOSBIFSBZCJSËODFLJBZEBOLPM- BJUDBOMMBSOTBZTIFSZMBSUNBLUBES UVLEBIBGB[MBZBQMZPS 1. ZMOEBBEBEBLJYUÑSÑOÑOTBZTLBÀUS \" # $ % & 5. ZMO TPOVOEB UPQMBN LBÀ LPMUVL ÑSFUJMNJö PMVS \" # $ 2. &OFSLFOIBOHJZMEBYUÑSÑOFBJUDBOMMBSOTBZ- % & TJHFÀNJöPMVS f f In_ 1, 5 i p c 4, 254 p In (1, 1) \" # $ % & WF TPSVMBS BøBôEBLJ CJMHJMFSF HÌSF 6. ,BÀBZTPOSBUPQMBNLPMUVLÑSFUJMNJöPMVS DFWBQMBZO[ \" # $ % & #JSGVUCPMTUBEZVNVOEBCËMÐNWBSES)FSCËMÐN- EF UPQMBN TSB WBSES )FS CËMÐN TSBEB LPMUVLTSBEBLPMUVLPMBDBLõFLJMEFBSUBSBL TSBZBLBEBSEFWBNFUNFLUFEJS 3. #JSCÌMÑNEFLBÀLPMUVLCVMVONBLUBES 7. ±[MFN PóMV 5PQSBL J¿JO EPóEVóVOEB CBOLBZB CJS \" # $ IFTBQB¿USZPSWFCVIFTBCB5-ZBUSZPS )FS¿FZSFLZMEBCJSPSBOOEBCJMFõJLGBJ[MFQB- % & SBZJõMFUNFZFCBõMZPS #VOB HÌSF 5PQSBL ZBöO CJUJSEJôJOEF CBO- LBEBLBÀ5-QBSBTPMVS 4. 4UBEZVNEBUPQMBNLBÀLPMUVLCVMVONBLUBES \" # \" # $ $ % % & & 1. A 2. C 3. D 4. B 34 5. B 6. C 7. B

%J[JMFS KARMA TEST - 1 1. #BZ#JMNJõËZMFCJSEJ[JZBQNõUSLJOZFSJOFEFO 5. UFSJNJWFJMLUFSJNUPQMBNPMBOBSJU- CBõMBZBO QP[JUJG UBN TBZMBSEBO IBOHJTJOJ LPZBSTB NFUJLEJ[JOJOUFSJNJLBÀUS LPZTVOTPOV¿BZO¿LNBLUBES \" # $ % & #BZ#JMNJöJOZBQUôEJ[JOJOHFOFMUFSJNJ 6. ^anh = 5n + 12 xn + 24 - 3n PMEVôVOBHÌSF YLBÀUS \" # $ % & 2. (FOFMUFSJNJ Z _ –1 in + 1 + sin_ nπ i , n çift ise :VLBSEBJ¿J¿FBMUHFOMFSõFLMJOEFBóËSFOCJSËSÐN- ]] DFL HËTUFSJMNJõUJS #V BMUHFOMFSJO ¿FWSF V[VOMVLMB- an = [ SCJSBSJUNFUJLEJ[JPMVõUVSNBLUBES&OLÐ¿ÐLJLJBM- ]] 4n , n tek ise UHFOJO¿FWSFV[VOMVLMBSCSWFCSEJS bn + c \\ \"ZSDBIFSCJSBSEöLBMUHFOJOCJSCJSMFSJOFCBô- MBOBOLÌöFMFSJBSBTV[BLMLCSPMEVôVOBHÌSF olan (an EJ[JTJOJO TBCJU EJ[J PMNBT JÀJO b + c CVÌSÑNDFLBMUHFOJUBNBNMBEôOEBUPQMBN LBÀPMNBMES LBÀCSBôÌSNÑöPMVS \" - # - $ - % - & - \" # $ % & 3. F = ' =WF'O+ = FO + FO- 7. #JSHFPNFUSJLEJ[JOJOJMLUFSJNÀBSQNOOJML öFLMJOEFUBONMBOBO'n'JCPOBDDJEJ[JTJOJO UFSJNÀBSQNOBPSBOPMEVôVOBHÌSF EJ[J- UFSJNJLBÀUS OJOUFSJNJLBÀUS \" # $ % & \" # $ % & 4. BO CJSBSJUNFUJLEJ[JEJS 8. BO CJSHFPNFUSJLEJ[JEJS a8 =B ve a = a + a - a = PMEVôVOBHÌSF CVEJ[JOJOJMLUFSJNJLBÀUS a - a = PMEVôVOBHÌSF CVEJ[JOJOPSUBLÀBSQBOLBÀUS \" # 5 $ % 7 & \" 1 # 2 $ 4 % 5 & 2 2 3 3 3 3 1. & 2. B 3. C 4. A 35 5. D 6. A 7. A 8. B

KARMA TEST - 2 %J[JMFS 1. AnTPOMVLÑNFTJOEFUBONM Bn EJ[JTJOJOHFOFM 4. #JSNBóB[BIFSHÐOCJSËODFLJHÐOTBUUóNBMMBSO UFSJNJ LBULBEBSTBUõZBQNBLUBES 3n - 9 .BôB[BJMLHÑOBEFUNBMTBUUôOBHÌSF n2 - 2n - 24 HÑOÑOTPOVOEBUPQMBNLBÀBEFUNBMTBUNöUS PMEVôVOBHÌSe, n en çoLLBÀPMBCJMJS \" - # - $ - 8 \" # $ % & % - & - 5. _ an i = 4n + 3 9 + xn - 3n EJ[JTJTBCJUEJ[JCFMJSUNFLUFEJS #VOBHÌSF YB1907BöBôEBLJMFSEFOIBOHJTJEJS \" # $ % & 2. ·ÀHFOTFMTBZEJ[JTJOJOHFOFMUFSJNJ 6. ÷MLUFSJNJWFPSUBLGBSLEPMBOCJSBSJUNFUJLEJ- n·^n+ 1h [JOJO WFUFSJNMFSJCJSHFPNFUSJLEJ[JPMVö- BO = f 2 p UVSEVôVOBHÌSF ELBÀUS Eá PMNBLÑ[FSF Bn ÑÀHFOTFMEJ[JTJOEF ( 3p - UFSJN Q+ 1 ). UFSJNJOLBUPMEVôVOB HÌSF QLBÀUS \" # $ % & \" - 16 # - 11 $ - % 1 & 11 33 33 7. +++++ UPQMBNOO ! TFNCPMÑ JMF HÌTUFSJNJ BöBôEBLJ- MFSEFOIBOHJTJPMBNB[ 3. #JSBQBSUNBOOIFSLBUOOZÐLTFLMJóJFSNFUSFEJS /20 /20 \"QBSUNBOO BTBOTËSÐ [FNJOEFO CBõMBNBL Ð[FSF \" 5k - 1 # _ 5k - 20 i TSBTZMBIFSLBUB¿LQ[FNJOFJOJZPS k =1 k =1 \"TBOTÌS LBUB ÀLUôOEB BMEô UPQMBN ZPM LBÀNFUSFPMVS 20 19 /$ _ 5k - 1 i /% _ 5k + 4 i k =1 k=0 \" # $ /18 % & & _ 5k + 9 i k =-1 1. B 2. C 3. & 36 4. D 5. A 6. A 7. A

%J[JMFS KARMA TEST - 3 1. #JMHJ(FOFMUFSJNJOp O- PMBOTBZEJ[JTJOFBM- 5. a <CPMNBLÐ[FSFBJMFCBSBTOBBSJUNFUJLEJ[JPMVõ- UHFOTFMTBZEJ[JTJEFOJS UVSBDBLõFLJMEFUFSJNZFSMFõUJSJMEJóJOEFPSUBLGBSL PMNBLUBES ( an BMUHFOTFM CJS TBZ EJ[JTJ PMNBL Ñ[FSF a4 = m · a8LPöVMVOVTBôMBZBONEFôFSJLBÀUS #VOBHÌSF a ile b OJOBSBTOBBSJUNFUJLEJ[J \" 30 # $ 3 % 7 & 7 22 7 7 15 30 PMVöUVSBDBL öFLJMEF UFSJN ZFSMFöUJSJMJSTF PMV- öBOEJ[JOJOPSUBLGBSLLBÀUS \" 18 # $ 21 % & 28 55 5 2. 4PZVUÐLFONFLUFPMBOCJSIBZWBOOZMOEBLJ 6. BO BSJUNFUJL EJ[JTJOJO BSEõL UFSJNJ N + QPQÐMBTZPOVEÐS O- Q+ CO BSJUNFUJLEJ[JTJOJOBSEõLUFSJ- NJN- O+Y Q+PMEVôVOBHÌSF YLBÀUS 1PQÑMBTZPO IFS ZM B[BMEôOB HÌSF JML IBOHJZMEBEFOEBIBB[IBZWBOLBMNöPMVS \" 5 # 2 $ % - & - 5 MPH à PMBSBLBMO[ 2 5 2 \" # $ % & 3. öOEJSHFNFCBóOUTOã GO+ =GO +GO-PMBO 7. Y Z [BSUBOCJSHFPNFUSJLEJ[JOJOBSEöLÑÀUFSJ- 'JCPOBDDJEJ[JTJOEFG13 = WFG12 = 144 PMEV- NJPMNBLÑ[FSF ôVOBHÌSF, f10LBÀUS Y+Z+[= \" # $ % & 1 + 1 + 1 = 13 x y z 18 PMEVôVOBHÌSF ZLBÀUS \" # $ % & 4. #JS TPOMV EJ[JEF Y EöOEBLJ UFSJNMFSJO UPQMB- 8. ( an CJSHFPNFUSJLEJ[JWF N ZEöOEBLJUFSJNMFSJOUPQMBN YWFZ B + B =BpB EöOEBLJ UFSJNMFSJO UPQMBN PMEVôVOB HÌSF a TPOMVEJ[JOJOUFSJNMFSJUPQMBNLBÀUS oMEVôVOBHÌSF 48 PSBOLBÀUS a 44 \" # $ \" # $ % & % & 1. & 2. D 3. D 4. & 37 5. C 6. & 7. C 8. C

KARMA TEST - 4 %J[JMFS 1. #JMHJ #JS HFPNFUSJL EJ[JOJO JML O UFSJNJOJO UPQMBN 5. öMLÐ¿UFSJNJY- Y+ Y -PMBOQP[JUJGUFSJN- 4O JMLOUFSJNJOJO¿BSQN1OWFJMLOUFSJNJOJO¿BS- MJ BO HFPNFUSJLEJ[JTJWFSJMJZPS CO BSJUNFUJLEJ[J- TJJ¿JOB =C ve a =CUÐS QNTBMUFSTMFSJOJOUPQMBN3OPMNBLÐ[FSF DO HFPNFUSJLEJ[JTJJ¿JOD =CWFD = aPMEV- f Sn n óVOBHËSF b7 JGBEFTJOJOEFôFSJLBÀUS Rn p = Pn2 EJS c2 ÷MLÑÀUFSJNJOJOBSJUNFUJLPSUBMBNBT HFPNFU- \" # $ % & SJLPSUBMBNBTPMEVôVOBHÌSF ÀBSQNTBMUFST- MFSJOJOUPQMBNLBÀUS \" # 7 $ 7 % 1 & 1 3 6 37 /2. n k - 1 - k - 2 = 5 k = 3 k2 - 3k + 2 6 n - 1 EJS 6. BO EJ[JTJOJOHFOFMUFSJNJBO = n! PMEVôVOBHÌSF OLBÀUS /18 \" # $ % & #VOBHÌSF ak UPQMBNOOEFôFSJLBÀUS k =1 \" 1- 1 # - 1 17! 18! $ - 1 % 1 - 1 17! 17! 18! & - 1 3. Y+ZáWFZáPMNBLÐ[FSF Y+Z Y +Z 18! Y +ZEJ[JTJIFNBSJUNFUJLIFNEFHFPNFUSJLCJS EJ[JOJOBSEõLUFSJNJPMEVóVOBHËSF YLBÀUS \" # $ & - % - 4. (FOFMUFSJNJ 7. #JSNFSEJWFOJOCBTBNBLMBSOOZÐLTFLMJLMFSJBSJUNF- BO =MPHf n + 1 p UJL EJ[J PMVõUVSBDBL õFLJMEF BZBSMBONõUS CBTB- n+2 NBL ZFSEFO DN CBTBNBL JTF ZFSEFO DNZVLBSEBES olan anEJ[JTJOJOJMLUFSJNJOJOUPQMBNLBÀUS &OÑTUCBTBNBôOZFSEFOZÑLTFLMJôJDNPM- \" - # MPH- $ EVôVOBHÌSFNFSEJWFOLBÀCBTBNBLMES % & MPH- \" # $ % & 1. C 2. C 3. B 4. B 38 5. D 6. & 7. A

%J[JMFS KARMA TEST - 5 1. D>WFQ R> 4. O>PMNBLÐ[FSFJMLUFSJNJCJS¿JGUUBNTBZWFPSUBL VO CJSBSJUNFUJLEJ[JPMNBLÐ[FSF GBSLPMBOCJSBSJUNFUJLEJ[JOJOJMLOUFSJNJOJOUPQMB- NPMEVóVOBHËSF OOJOBMBCJMFDFôJEFôFSMFS V =MPHDQWFV =MPHD QpR EJS UPQMBNLBÀUS 16 /Q=DWFR=DPMEVóVOBHËSF u UPQMBN- k \" # $ % & OOEFôFSJLBÀUS k =1 \" # $ % & 5. _ an + 1 i · 1+ 4_ an i2 = 1 2. BO TOSMEJ[JTJOJOUFSJNMFSJ an a = a = WFB1 =PMBDBLöFLJMEFLJQP[JUJGUFSJNli ( an EJ- a = [JTJJÀJOa31 LBÀUS \" # $ % 1 & 7 aO = PMNBLÐ[FSF 6. #JSNÐ[JLTUÐEZPTVOEBTFTLBZUMBSOOEBIBUFNJ[ OUBOF PMNBTJ¿JOTUÐEZPOVOEVWBSMBSEõBSEBOHFMFCJMF- anEJ[JTJOJOHFOFMUFSJNJBöBôEBLJMFSEFOIBOHJ- DFLTFTJPSBOOEBHF¿JSFOCJSNBEEFJMFLBQ- TJEJS MBOZPS \" 9n - 1 # 10n - 1 4FTJO PSBOOEB HFÀFCJMNFTJ JÀJO CV NBE- 10 9 EFEFOLBÀLBUTÑSÑMNFMJEJS $ 10n - 1 % 9n - 1 # In2 - 1 39 In6 & 10n - 9n \" *O $ MPH 3 % MPH log 2 - 1 & log 6 - 1 3. 7. #JMHJ #JSEJ[JOJOUFSJNMFSJOFBJUËSÐOUÐZVLBSEBLJHJCJEJS n 1 11 1n #VOBHËSF / = + + ...+ = a = k = 1 k^ k + 1 h 1· 2 2 · 3 n^ n + 1 h n + 1 a = a = 8 EJS a = PMBDBLõFLJMEFËSÐOUÐEFWBNFUUJSJMNFLUFEJS #VCJMHJZFHÌSF %J[JOJO UÑN UFSJNMFSJ JÀJO CV ÌSÑOUÑ HFÀFSMJ PM- 5 + 13 + 25 + . . . + 841 EVôVOBHÌSF B8 + a13UPQMBNLBÀUS 4 12 24 840 \" # $ % & UPQMBNOOEFôFSJLBÀUS \" # 449 $ 430 20 21 % 349 & 300 21 7 1. A 2. C 3. & 39 4. B 5. & 6. & 7. C

<(1m1(6m/6258/$5 %J[JMFS WF TPSVMBS BøBôEBLJ CJMHJMFSF HÌSF WF TPSVMBS BøBôEBLJ CJMHJMFSF HÌSF DFWBQMBZO[ DFWBQMBZO[ #JSB¿LIBWBUJZBUSPTVOEBPUVSNBEÐ[FOJOFBJUCJM- öLJCËDFLUÐSÐWFCVUÐSMFSFBJUDBOMTBZMBSJMFJMHJ- HJMFSBõBóEBLJHJCJWFSJMNJõUJS MJZBQMBOCJSBSBõUSNBEBBõBóEBLJWFSJMFSUPQMBO- r OTSBEBLJLPMUVLTBZTO+TSBEBLJLPMUVL ZPS TBZTOEBOFLTJLUJS r TSBEBLPMUVLCVMVONBLUBES 5·3\" 5·3# r 4POTSBEBLJLPMUVLTBZTUS ASBõUSNBZBCBõMBOE- \"SBõUSNBZBCBõMBOE- 1. 4BMPOEBLBÀTSBLPMUVLWBSES óOEBUÐSÐOQPQÐMBT- óOEBUÐSÐOQPQÐMBT- ZPOVEJS ZPOVEJS \" # $ % & 5ÐSÐOTBZTOEBIFS 5ÐSÐOTBZTOEBIFS BZMJLBSUõIFTBQ- BZBEFUB[BMNB MBONõUS HËSÐMNÐõUÐS 4. \"UÑSÑOÑOTBZTOOCJOJHFÀUJôJOJOHÌ[MFN- MFOFCJMNFTJJÀJOBSBöUSNBFOB[LBÀBZTÑSNFMJ- EJS f ln 1, 1 . 4, 81 p ln 1, 02 \" # $ % & 2. 5JZBUSPZBCJSTFGFSEFFOÀPLLBÀLJöJHFMFCJMJS \" # $ 5. # UÐSÐOÐO QPQÐMBTZPOVOEBLJ B[BMNB HË[MFNMFOF- % & SFL JML TBZNEBLJ QPQÐMBTZPOVOVO VOVO BMU- OB EÐõNFTJ IBMJOEF UÐSÐO LPSVNB BMUOB BMONBT- OBLBSBSWFSJMJZPS #VOBHÌSF BSBöUSNB UÑSLPSVNBBMUOBBMONB- EBOÌODFCBöMBOHÀUBOJUJCBSFOIBOHJZMMBSBSB- TOEBUBNBNMBOS \" - # - $ - % - & - 3. %Ñ[FOMFOFDFLCJSHÌTUFSJEFFOÌOEFOFOBSLB- 6. \" UÑSÑOÑO QPQÑMBTZPOVOVO # UÑSÑOÑO QPQÑ- ZBEPôSVFOB[LBÀTSBCJMFUTBUMSTBTBUMBCJMF- MBTZPOVOEBOGB[MBPMEVôVJMLHÌ[MFNMFOEJôJOEF DFLCJMFUMFSJOZBSTOEBOGB[MBTTBUMNöPMVS BSBöUSNBZBCBöMBZBMLBÀBZPMNVöUVS à \" # $ % & \" # $ % & 1. B 2. A 3. B 40 4. D 5. A 6. C

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AYT Matematik Ders İşleyiş Modülleri 4. Modül Diziler

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AYT Matematik Ders İşleyiş Modülleri 4. Modül Diziler

Description: AYT Matematik Ders İşleyiş Modülleri 4. Modül Diziler

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