56385272 Δομή Επανάληψης Ασκήσεις Επανάληψης Πριν Τις...

8/19/2019 56385272 Δομή Επανάληψης Ασκήσεις Επανάληψης Πριν Τις Εξετάσεις

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Na anaptucqeί algόriqμoV pou qa ektupώnei touV diyήfiouV άrtiouV akέraiouV.EP.1AlgόriqμoV άrtioi_diyήfioiTέloV άrtioi_diyήfioiH άskhsh sthn ousίa qa prέpei na ektupώsei tou ariqμoύV 10, 12, 14, ¼.,96, 98.Me μia prώth μatiά qa μporoύsaμe na thn lύsouμe μe to na prospelάsouμe tiV tiμέV όlwn tdiyήfiwn kai μe th crήsh doμήV epilogήV na epilέxouμe touV άrtiouV.O algόriqμoV eίnai o akόlouqoV:

apό 10 Gia! diyήfioiμέcri 99An m 2=0 tόteEktύpwse

TέloV_anTέloV_epanάlhyhVAn to grάyw μe autό ton trόpo:AlgόriqμoV άrtioi_diyήfioiTέloV άrtioi_diyήfioi

apό 10 Gia! qa μporoύse na anέbei kai μέcri to 99μέcri 98Ektύpwse TέloV_epanάlhyhV

μe_bήμa 2ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEPANALHYHO taμίaV tou m katacwreί sthn taμeiakή apόdeixh ton kwdikό tou pro

thn tiμή. Όtan oloklhrwqeί h katacώrhsh gia thn agorά enόV pelάth , tόte eisάgetai o kwanaptύxete algόriqμo pou qa ulopoieί thn parapάnw diadikasίa kai sto tέloV qa ektupώsunolikό posό thV agorάV kai to plήqoV twn proϊόntwn pou agorάsthkan.Όtan to plήqoV twn epanalήyewn eίnai άgnwsto, tόte den μporoύμe na crhsiμopoiήsouμe th epanάlhyhV Gia¼.Sthn perίptwsh autή crhsiμopoioύμe thn Όso (ή th MέcriV_όtou).1o bήμa: Δiatupώnw th sunqήkh pou epiquμoύμekwdikόV0 Όso epanέlabeTέloV_epanάlhyhV2o

bήμa: Arcikopoiώ th μetablhtήΔiάbase kwdikόV3o bήμa: Topoqetώ thn entolή allagήVΔiάbase kwdikόVOpόte ftiάxaμe to basikό skeletό thV epanάlhyhV.Sth sunέceia suμplhrώnouμe tiV upόloipeV entolέV.Arcikopoίhsh sunόlou prin to brόco.Aύxhsh (μetabolή) sunόlou μέsa sto brόco.


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sύnolo ¬0Δiάbase tiμή , teμάciasύnolo ¬ sύnolo + tiμή*teμάciaMaV zhteίtai parάllhla kai to plήqoV twn proϊόntwn .plήqoV ¬0Arcikopoίhsh plήqoV.plήqoV¬ plήqoV+ 1Metabolή thV μetablhtήV plήqoV(Άύxhsh).ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.2 EPANALHYHO taμίaV tou

m

katacwreί sthn taμeiakή apόdeixh ton kwdikό tou prothn tiμή. Όtan oloklhrwqeί h katacώrhsh gia thn agorά enόV pelάth , tόte eisάgetai o kwanaptύxete algόriqμo pou qa ulopoieί thn parapάnw diadikasίa kai sto tέloV qa ektupώsunolikό posό thV agorάV kai to plήqoV twn proϊόntwn pou agorάsthkan.O algόriqμoV eίnai o akόlouqoV:

AlgόriqμoV

TέloV

kwdikόV0 Όso epanέlabeTέloV_epanάlhyhVΔiάbase kwdikόVΔiάbase kwdikόVsύnolo ¬0Δiάbase tiμή , teμάciasύnolo ¬ sύnolo + tiμή*teμάciaplήqoV ¬0plήqoV¬ plήqoV+ 1Ektύpwse sύnolo , plήqoV

Arcήsύnolo¬0plήqoV¬0ΔiάbasekwdikόVbrόcoVkwdikόV 0NaiΔiάbasetiμή,teμάciasύnolo ¬ sύnolo +tiμή*teμάciaplήqoV ¬ plήqoV+1Δiάbase

kwdikόVΌciEktύpwsesύnolo ,plήqoVTέloVANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.2 EPA


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NALHYHO algόriqμoV eίnai o akόlouqoV:AlgόriqμoV perittoί_μoTέloV perittoί_μoArcή_epanάlhyhVάqroisμa ¬0Δiάbase ariqμόVplήqoV ¬0Na anaptucqeί algόriqμoV pou qa diabάzei akέraiouV ariqμoύV apό ton crήsth ,qaupologίzei kai qa ektupώnei ton μέso όro twn perittώn ap' autoύV touV ariqμoύV .H diadikasίa anάgnwshV qa staμatά όtan έcoun diabasteί 20 perittoί ariqμoί.O ariqμόV twn epanalήyewn eίnai άgnwstoV.H diatύpwsh thV ekfώnhshV μaV parapέμpei sthcrήsh thV doμήV epanάlhyhV MέcriV_όtou .Me thn Όso¼ μporeί na μhn gίnei potέ epanάlhyh.Me th MέcriV_όtou toulάciston μίa.An ariqμoV m

2=1 tόteάqroisμa ¬ άqroisμa +ariqμόVplήqoV ¬ plήqoV +1

TέloV_anMέcriV_όtou plήqoV=20An plήqoV 0 tόteμέsoV_όroV ¬ άqroisμa /plήqoVEktύpwse μέsoV_όroVTέloV_anANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.3 EPANA

LHYHO algόriqμoV eίnai o akόlouqoV:AlgόriqμoV perittoί_μoTέloV perittoί_μoArcή_epanάlhyhVάqroisμa ¬0Δiάbase ariqμόVplήqoV ¬0An ariqμoV m

2=1 tόteάqroisμa ¬ άqroisμa +ariqμόV

plήqoV ¬ plήqoV +1TέloV_anMέcriV_όtou plήqoV=20An plήqoV 0 tόteμέsoV_όroV ¬ άqroisμa /plήqoVEktύpwse μέsoV_όroVTέloV_anArcήάqroisμa¬0plήqoV¬0


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ΔiάbaseariqμόVariqμόVm 2 =1Naiάqroisμa ¬ άqroisμa+ariqμόVplήqoV ¬ plήqoV+1plήqoV=20ΌciΌciNai TέloV brόcouplήqoV 0NaiμέsoV_όroV ¬ άqroisμa/plήqoVEktύpwseμέsoV_όroVTέloVΌcibrόcoVANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.3 EPA

NALHYHNa anaptucqeί algόriqμoV pou qa upologίzei kai qa ektupώnei to άqroisμa twn qetikώndiyήfiwn perittώn akeraίwn .Δhl to άqroisμa : =11+13+15+¼.+95+97+99AlgόriqμoV Aqroisμa_PerittώnTέloV Aqroisμa_PerittώnGia thn epίlush thV άskhshV qa crhsiμopoiήsouμe έnan aqroistή. ΈnaV aqroistήV periέcei dύo entolέV:

a. Mhdenisμό prin thn epanάlhyhb. Aύxhsh tou aqroistή entόV tou brόcouO algόriqμoV eίnai oakόlouqoV: ¬0 apό 11 Gia μέcri 99 μe_bήμa 2

¬

+

TέloV_epanάlhyhVEktύpwse ªTo άqroisμa eίnai º, ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.4 E

PANALHYHNa anaptucqeί algόriqμoV pou qa upologίzei kai qa ektupώnei ta aqroίsμata:


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1= 1 +12+13+14+¼.+1100 kai

2=13+19+127

+¼.+13100StiV askήseiV autέV prέpei na xecwrίsouμe poio eίnai to stoiceίo tou aqroίsμatoV pou alpoio paraμέnei aμetάblhto.

1=11+1

2+13+14+¼.+1100Sto prώto άqroisμa o ariqμhtήV paraμέnei staqerόV.Enώ o paronoμastήV μetabάlletai kai paίrnei diadocikά tiV tiμέV 1,2,3,4,5,6¼,99,100.

2=131+132+


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133+¼.+13100Sto deύtero άqroisμa o ariqμhtήV paraμέnei staqerόV.Enώ ston paronoμastή upάrcei μia dύnaμh tou 3 kai o ekqέthV μetabάlletai kai paίrnei ditiμέV 1,2,3,4,5,6¼,99,100.ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.5 EPANALHYHNa anaptucqeί algόriqμoV pou qa upologίzei kai qa ektupώnei ta aqroίsμata:

1

= 1 +12+13+14+¼.+1100 kai

2=13+19+127+¼.+13

100

1=11+12+


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13+14+¼.+1100

2=131+132+133+¼.+1

3100AlgόriqμoV ΆqroisμaTέloV ΆqroisμaO algόriqμoV eίnai oakόlouqoV: _1¬0 _2¬0 apό 1 Gia μέcri 100 _1¬ _1 +1 / _2¬ _2 +1 /3^ TέloV_epanάlhyhV

Ektύpwse ªTa aqroίsμata eίnai º,

_1,

_2ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.5 EPANALHYHNa anaptucqeί algόriqμoV pou qa diabάzei έnan akέraio ariqμό N kai qa upologίzei to

άqroisμa:

1= 5 + 3 - 9 +27-81+¼ +-


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3NSto άqroisμa autό parathroύμe όti ,μetά ton arcikό όro 5,to prόshμo kaqenόV όrou enallάssetai.Thn enallagή tou prosήμou μporoύμe na thn epitύcouμe μe thn ύywsh tou -1 se άrtio ή pekqέth, anάloga μe thn exέlixh twn tiμώn tou μetrhtή thV doμήV epanάlhyhV.Όtan to (-1) uyώnetai se perittό ekqέth to prόshμo gίnetai arnhtikό,enώ όtan uyώnetai se άrtio ekqέth gίnetai qetikό.To enallassόμeno prόshμo dhμiourgeίtai pollaplasiάzontaV μe ton όro (-1) ̂ ή ton (-1)^(-1)^1+1 (-1)^2+1 (-1)^3+1 ¼¼¼..AlgόriqμoV Aqroisμa_PrόshμoTέloV Aqroisμa_PrόshμoO algόriqμoV eίnai oakόlouqoV:Grάye ªΔώse έnan akέraio ariqμό ºΔiάbase N apό 1 Gia μέcri N ¬5

¬

+(-1)^(1+

)*(3)̂

TέloV_epanάlhyhVGrάye ªH tiμή tou aqroίsμatoV º,

ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhV

EP.6 EPANALHYHNa anaptucqeί algόriqμoV pou qa diabάzei έnan akέraio ariqμό N kai qa upologίzei toάqroisμa: =2 + 4 +

8+16+¼ +2N1 +32+94

+276+¼.+3N2*N, an N άrtioV, an N perittόVAlgόriqμoV Δiplό_Άqroisμa


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TέloV Δiplό_ΆqroisμaO algόriqμoV eίnai oakόlouqoV:Grάye ªΔώse έnan akέraio ariqμό ºΔiάbase NAn N m

2=0 tόte ¬0

apό 1 Gia μέcri N

¬

+ 2̂

TέloV_epanάlhyhVAlliώV ¬1

apό 1 Gia μέcri N

¬

+ 3̂

/(2*

)TέloV_epanάlhyhVTέloV_anGrάye ªH tiμή tou aqroίsμatoV º, ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.7 EPA

NALHYHTo paragontikό orίzetai wV exήV:N! =aόristo, an N0

Na anaptucqeί algόriqμoV pou qa diabάzei έnan ariqμό kai qa upologίzei to paragontikόtou. AlgόriqμoV ParagontikόTέloV ParagontikόO algόriqμoV eίnai oakόlouqoV:Grάye ªΔώse έnan akέraio ariqμό ºΔiάbase NAn N < 0 tόteGrάye ªTo paragontikό den orίzetaiºAlliώV_an N = 0 tόteGrάye ªTo 0 paragontikό eίnai 1ºAlliώV

ginόμeno ¬1 apό 1 Gia μέcri Nginόμeno¬ ginόμeno* TέloV_epanάlhyhV! Oudέtero stoiceίo tou pollaplasiasμoύGrάye ªToº, N ª paragontikό eίnai º , ginόμenoTέloV_anANAPTUXH EFARMOGΩN∆oμή EpanάlhyhV


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EP.8 EPANALHYHNa anaptucqeί algόriqμoV pou qa ektupώnei tiV lύseiV thV exίswshV5

+12

-6

=11μe ta

,

,

na paίrnoun tiV akέraieV tiμέV [-50,50].AlgόriqμoV ExίswshTέloV ExίswshGia thn epίlush thV άskhshV qa prέpei oi μetablhtέV

,

,

, na pάroun όleV tiV epitreGia kάqe μίa tiμή prέpei na upologίzetai to άqroisμa5

+12

-6

thV ekfώnhshV kai na elέgcetai an to apotέlesμa isoύtai μe 11, opόte oi tiμέV twn

,

apoteloύn lύsh thV exίswshV.Gia autό qa crhsiμopoiήsouμe 3 eμfwleuμέneV doμέV epanάlhyhV Gia...Prosocή:Prέpei na oloklhrώnetai kάqe eswterikόV brόcoV protoύ sunecisteί o exwterikόV.O algόriqμoV eίnai oakόlouqoV:

apό -50 Gia

μέcri 50

apό -50 Gia μέcri 50 apό -50 Gia μέcri 50posόthta ¬ 5*

+12*

-6*

An posόthta=11 tόteEktύpwse ºLύsh thV exίswshV eίnai :º

,

,

TέloV_anTέloV_epanάlhyhVTέloV_epanάlhyhVTέloV_epanάlhyhV

ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.9 EPANALHYHNa anaptucqeί algόriqμoV pou qa ektupώnei touV 200 prώtouV όrouV thV akolouqίaV:a

n=22*an-1+(n-1), an n=1, an n>1AlgόriqμoV Akolouqίa_anadroμikήTέloV Akolouqίa_anadroμikή


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Se kάqe epanάlhyh , prέpei ston prohgoύμeno όro na prosqέtouμe thn epiplέon posόthta n-O algόriqμoV eίnai oakόlouqoV: a_n ¬ 2Grάye ºO 1oV όroV thV akolouqίaV eίnai :ºa_n! Ektύpwsh zhtά h άskhsh. apό 2 Gia μέcri 200a_n ¬ 2* a_n +( -1)Grάye ºO º,

,ªoV όroV thV akolouqίaV eίnai :ºa_nTέloV_epanάlhyhV! Mhn se μperdeύei to an-1 thV ekfώnhshVANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.10 EPANAL

HYHNa anaptucqeί algόriqμoV pou diabάzei έnan ariqμό N>2 kai qa ektupώnei antίstoicoplήqoV όrwn thV akolouqίaV . O prώtoV όroV thV akolouqίaV eίnai to 0,odeύteroV to 1 kai kάqe epόμenoV όroV eίnai to άqroisμa twn dύo prohgouμέnwn( 0, 1 ,1 , 2, 3, 5 ,8 , 13 , 21 ,34 , 55 , 89,¼.)Se autή thn άskhsh ta prάgμata periplέkontai . Gia ton upologisμό tou ekάstote όrou creprohgoύμenoi dύo όroi thV akolouqίaV. Qa crhsiμopoiήsouμe dύo μetablhtέV pou qa diathroprohgoύμeno kai ton pro-prohgoύμeno όro antίstoica.AlgόriqμoV Akolouqίa_ TέloV Akolouqίa_ O algόriqμoV eίnai o

akόlouqoV: Δiάbase N ! Qewroύμe όti N>2Grάye ºO 1oV όroV thV akolouqίaV eίnai 0ºGrάye ºO 2oV όroV thV akolouqίaV eίnai 1ºa_n_2 ¬ 0a_n_1 ¬ 1 apό 3 Gia μέcri Na_n ¬ a_n_2 + a_n_1

Grάye ºO º,

,ªoV όroV thV akolouqίaV eίnai :ºa_na_n_2 ¬ a_n_1 ! pro-prohgoύμenoV όroV o μέcri tώra prohgoύμenoVa_n_1 ¬ a_n! prohgoύμenoV όroV o trέcwnTέloV_epanάlhyhV! an-2pro-prohgoύμenoV όroV! an-1


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prohgoύμenoV όroV! an-2+ an-1ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.11 EPANALHYHNa anaptucqeί algόriqμoV pou qa ektupώnei touV qetikoύV triyήfiouV akέraiouV poueίnai pollaplάsioi tou 7, kaqώV kai to plήqoV twn ariqμώn autώn.Qa crhsiμopoiήsouμe έna μetrhtή pou qa kataμetrά touV triyήfiouV akέraiouV pou eίnaipollaplάsioi tou 7.O μetrhtήV apaiteί 2 entolέV :1. Mhdenisμό tou μetrhtή prin epanάlhyh.2. Aύxhsh thV tiμήV katά μίa μonάda , entόV tou brόcou.AlgόriqμoV Pollaplάsia_7

TέloV Pollaplάsia_7O algόriqμoV eίnai oakόlouqoV: plήqoV ¬ 0 apό 100 Gia μέcri 999An m 7=0 tόte ! Pollaplάsio tou 7plήqoV ¬ plήqoV+1Grάye ºO ariqμόV ª, , ªikanopoieί tiV proϋpoqέseiVºTέloV_anTέloV_epanάlhyhVGrάye plήqoV

ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.12 EPANALHYHNa anaptucqeί algόriqμoV pou qa ektupώnei touV qetikoύV triyήfiouV akέraiouV poueίnai pollaplάsioi tou 7, kaqώV kai to plήqoV twn ariqμώn autώn.

O algόriqμoV μporeί na grafeί kai μe άllo trόpo.Qa xekinήsouμe μe apό ton prώto triyήfio akέraio pou eίnai pollaplάsioV tou 7 ( ton 105bήμa 7 qa diatrέxouμe όla ta pollaplάsia, μέcri ton teleutaίo triyήfio.AlgόriqμoV Pollaplάsia_7TέloV Pollaplάsia_7O algόriqμoV eίnai oakόlouqoV: plήqoV ¬ 0 apό 105 Gia μέcri 999 μe_bήμa 7


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plήqoV ¬ plήqoV+1Grάye ºO ariqμόV ª,

, ªikanopoieί tiV proϋpoqέseiVºTέloV_epanάlhyhVGrάye plήqoVANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.12 EPANALHYHNa anaptucqeί algόriqμoV pou qa ektupώnei όlouV touV qetikoύV triyήfiouV akέraiouV,twn opoίwn h tiμή eίnai ίsh μe to άqroisμa twn kύbwn twn yhfίwn touV(Ariqμoί

m

).H άskhsh μporeί na epiluqeί μe dύo proseggίseiV.· H 1h prosέggish eίnai na diatrέxouμe όlouV touV triyήfiouV akέraiouV , na apoμonώsouμe ta 3yhfίa touV kai na elέgxouμe th sunqήkh thV ekfώnhshV.AlgόriqμoV kύboV_triyήfiwn

TέloV kύboV_triyήfiwnO algόriqμoV eίnai oakόlouqoV: C apό 100 Gia μέcri 999ekatontάdeV ¬ C v 100bohqhtikή ¬ C m 100dekάdeV ¬ bohqhtikή v 10μonάdeV ¬ bohqhtikή m 10posόthta ¬ ekatontάdeV^3+ dekάdeV^3+ μonάdeV^3An posόthta=C tόteGrάye ºO ariqμόV ª,C, ªikanopoieί tiV proϋpoqέseiVºTέloV_anTέloV_epanάlhyhV

ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.13 EPANALHYHNa anaptucqeί algόriqμoV pou qa ektupώnei όlouV touV qetikoύV triyήfiouV akέraiouV,twn opoίwn h tiμή eίnai ίsh μe to άqroisμa twn kύbwn twn yhfίwn touV(Ariqμoί

m

).H άskhsh μporeί na epiluqeί μe dύo proseggίseiV.· H 1h prosέggish eίnai na diatrέxouμe όlouV touV triyήfiouV akέraiouV , na apoμonώsouμe ta 3yhfίa touV kai na elέgxouμe th sunqήkh thV ekfώnhshV.· H 2h prosέggish eίnai μe th crήsh triώn doμώn epanάlhyhV , ώste na dhμiourgήsouμe tiVekatontάdeV, dekάdeV kai μonάdeV (tiμέV 0 έwV 9) dhladή trίa yhfίa enόV triyήfiou ariqμ


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elέgxouμe th sunqήkh thV ekfώnhshV.AlgόriqμoV kύboV_triyήfiwnTέloV kύboV_triyήfiwnO algόriqμoV eίnai oakόlouqoV: ekatontάdeV apό 1Gia μέcri 9 ! An άrcize apό 0 den qa ήtan triyήfioVdekάdeV apό 0Gia μέcri 9μonάdeV apό 0 Gia μέcri 9posόthta ¬ ekatontάdeV^3+ dekάdeV^3+ μonάdeV^3C ¬100*ekatontάdeV+ 10*dekάdeV+ μonάdeVAn posόthta =C tόteGrάye ºO ariqμόV ª,C, ªikanopoieί tiV proϋpoqέseiVºTέloV_anTέloV_epanάlhyhVTέloV_epanάlhyhVTέloV_epanάlhyhVANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.13 EPA

NALHYHNa anaptucqeί algόriqμoV pou qa brίskei kai qa ektupώnei όlouV touV qetikoύVtetrayήfiouV akέraiouV pou μporoύn na diabastoύn kai anάpoda (gia parάdeigμa 1331,7447, 9229 klp .).H άskhsh μporeί na epiluqeί μe diάspash twn yhfίwn tou ariqμoύ μe th diaforά όti qaantistoicήsouμe tiV ciliάdeV μe tiV μonάdeV kaitiV ekatontάdeV μe tiV dekάdeV .AlgόriqμoV tetrayήfioV_anάpoda

O algόriqμoV eίnai oakόlouqoV: C apό 1000 Gia μέcri 9999ciliάdeV ¬ C

v 1000bohqhtikή ¬ C m 1000ekatontάdeV ¬ bohqhtikή v 100bohqhtikή ¬ bohqhtikή m 100TέloV tetrayήfioV_anάpodadekάdeV ¬ bohqhtikή v 10μonάdeV ¬ bohqhtikή m 10An ciliάdeV =μonάdeV kai ekatontάdeV=dekάdeV tόteGrάye ºO ariqμόV ª,C, ªikanopoieί tiV proϋpoqέseiVº

TέloV_anTέloV_epanάlhyhVANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.14 EPANAL


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HYHNa anaptucqeί algόriqμoV pou qa diabάzei έna ariqμό N μegalύtero tou 1 kai qa elέgceian iscύei h isόthta:(1+2+3+¼.+N)2=13+23+33+43+¼..+N3AlgόriqμoV EpalήqeushO algόriqμoV eίnai oakόlouqoV:TέloV EpalήqeushΔiάbase Nάqroisμa1 ¬0άqroisμa2 ¬0

apό 1 Gia μέcri

! Qewrώ όti N>1άqroisμa1 ¬ άqroisμa1 + άqroisμa2 ¬ άqroisμa2 + ^3TέloV_epanάlhyhVάqroisμa1 ¬ άqroisμa1^2An άqroisμa1 = άqroisμa2 tόteGrάye ºH isόthta thV ekfώnhshV iscύei.ºAlliώVGrάye ºH isόthta thV ekfώnhshV den iscύei.ºTέloV_anANAPTUXH EFARMOGΩN

∆oμή EpanάlhyhVEP.15 EPANALHYHNa anaptucqeί algόriqμoV pou qa diabάzei 500 qetikoύV akέraiouV ariqμoύV kai qaektupώnei to posostό twn ariqμώn pou eίnai diyήfioi.AlgόriqμoV Posostό_Δiyήfiwn

O algόriqμoV eίnai oakόlouqoV:TέloV Posostό_Δiyήfiwndiyήfioi ¬0 apό 1 Gia μέcri 500Δiάbase ariqμόVAn ariqμόV >= 10 kai ariqμόV


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TέloV_epanάlhyhVposostό ¬100*diyήfioi/500Grάye ºTo posostό twn diyήfiwn eίnaiº,posostόANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.16 EPANALHYHNa anaptucqeί algόriqμoV pou qa diabάzei N akέraiouV ariqμoύV (όpou N dedoμέnoVakέraioV) kai qa ektupώnei to άqroisμa kai to μέso όro άrtiwn.Όtan kaloύμaste na upologίsouμe kάpoio μέso όro , prέpei na upologίsouμe ton antίstoicoariqμhtή kai paronoμastή.O paronoμastήV sunήqwV eίnai to plήqoV twn epanalήyewn ,enώ gia ton ariqμhtή arkeί h crήsh enόV aqroistή.AlgόriqμoV ΆrtioiO algόriqμoV eίnai oakόlouqoV:TέloV Άrtioi

Δedoμέna // N //άqroisμa_άrtiwn ¬0plήqoV_άrtiwn ¬0 apό 1 Gia μέcri Δiάbase ariqμόVAn ariqμόV m 2= 0 tόteάqroisμa_άrtiwn ¬ άqroisμa_άrtiwn +ariqμόVplήqoV_άrtiwn ¬ plήqoV_άrtiwn + 1TέloV_anTέloV_epanάlhyhVAn plήqoV_artίwn 0 tόte ! Kritήrio kaqoristikόthtaVμέsoV_όroV ¬ άqroisμa_άrtiwn / plήqoV_άrtiwn

Grάye άqroisμa_άrtiwn , μέsoV_όroVTέloV_anANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.17 EPANALHYH

Na anaptucqeί algόriqμoV pou qa diabάzei to polύ 100 akέraiouV ariqμoύV , apό touVopoίouV to polύ 20 qa eίnai arnhtikoί. O algόriqμoV qa ektupώnei to posostό twnqetikώn kai twn arnhtikώn ariqμώn pou diάbase (ta μhdenikά den qewroύntai qetikoί oύtearnhtikoί).Ftiάcnouμe ton skeletό thV epanάlhyhV:To polύ 100 akeraίouV.Όso (όloi


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arnhtikoί ¬0O algόriqμoV eίnai oakόlouqoV:AlgόriqμoV Posostά_ariqμώnTέloV Posostά_ariqμώnqetikoί ¬0Δiάbase ariqμόVAn ariqμόV < 0 tόtearnhtikoί ¬ arnhtikoί+1AlliώV_an ariqμόV > 0 tόteqetikoί ¬ qetikoί +1TέloV_anόloi ¬ όloi+1posostό_qetikώn ¬100*qetikoί/όloiposostό_arnhtikώn ¬100*arnhtikoί/όloiGrάye posostό_qetikώn ,posostό_arnhtikώnANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.18 EPA

NALHYHNa anaptucqeί algόriqμoV pou qa diabάzei ariqμoύV agnώstou plήqouV kai qa ektupώneito μέso όro touV. H epanalhptikή diadikasίa qa terμatίzetai όtan doqeί o ariqμόV -9999 opoίoV den qa kataμetrάtai.Qa epilύsouμe thn άskhsh μe th crήsh thV doμήV MέcriV_όtou.άqroisμa ¬0plήqoV ¬0Arcή_epanάlhyhV

Δiάbase ariqμόVάqroisμa ¬ άqroisμa +ariqμόVplήqoV ¬ plήqoV+ 1MέcriV _όtou ariqμόV =-9999O algόriqμoV έcei logikό lάqoV.O ariqμόV pou odhgeί ston terμatisμό qa prosteqeίkai autόV sto άqroisμa kai qa kataμetrhqeί sto plήqoV,kάti pou qa odhgήsei se lάqoV apotelέsμata.AutόV to lάqoV eίnai sunέpeia thV idiόthtaV thV doμήVMέcriV_όtou , όpou h sunqήkh elέgcetai sto tέloV toubrόcou kai , kai diorqώnetai μe th prosqήkh thVdoμήV epilogήV An.

ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.19 EPANALHY


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HNa anaptucqeί algόriqμoV pou qa diabάzei ariqμoύV agnώstou plήqouV kai qa ektupώneito μέso όro touV. H epanalhptikή diadikasίa qa terμatίzetai όtan doqeί o ariqμόV -9999 opoίoV den qa kataμetrάtai.AlgόriqμoV MO_AriqμώnO algόriqμoV eίnai o akόlouqoV:TέloV MO_Ariqμώnάqroisμa ¬0plήqoV ¬0Arcή_epanάlhyhVΔiάbase ariqμόVAn ariqμόV -9999 tόteάqroisμa ¬ άqroisμa +ariqμόVplήqoV ¬ plήqoV+ 1TέloV_anMέcriV _όtou ariqμόV =-9999An plήqoV 0 tόte ! Kritήrio kaqoristikόthtaVμέsoV_όroV ¬ άqroisμa/plήqoVGrάye plήqoV,μέsoV_όroVAlliώVGrάye ªΔen dόqhke kanέnaV ariqμόVº

TέloV_anParathroύμe όti upάrcei perίptwsh naμhn ektelesteί kaμίa forά to tμήμaentolώn tou brόcou ,άra μάllon tairiάzei h doμή epanάlhyhVΌso .ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.19 EPANA

LHYHNa anaptucqeί algόriqμoV pou qa diabάzei ariqμoύV agnώstou plήqouV kai qa ektupώneito μέso όro touV. H epanalhptikή diadikasίa qa terμatίzetai όtan doqeί o ariqμόV -9999 opoίoV den qa kataμetrάtai.AlgόriqμoV MO_AriqμώnO algόriqμoV eίnai o akόlouqoV:TέloV MO_Ariqμώnάqroisμa ¬0plήqoV ¬0Δiάbase ariqμόV

ΌsoariqμόV -9999 epanέlabeάqroisμa ¬ άqroisμa +ariqμόVplήqoV ¬ plήqoV+ 1TέloV_epanάlhyhVAn plήqoV 0 tόte ! Kritήrio kaqoristikόthtaVμέsoV_όroV ¬ άqroisμa/plήqoVGrάye plήqoV,μέsoV_όroVAlliώV


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Grάye ªΔen dόqhke kanέnaV ariqμόVºTέloV_anProstέqhke άllh μia entolήΔiάbase ,ώste na arcikopoihqeί h μetablhtήariqμόV.Δiάbase ariqμόVSth sunέceia , sto tέloV tou brόcoudiabάzetai h epόμenh tiμή gia nacrhsiμopoihqeί ston epόμeno έlegco.Proteίnetai h doμή Όso gia epίlushaskήsewn όpou to plήqoV twnepanalήyewn eίnai άgnwsto.ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.19 EPANALHYH

Na anaptucqeί algόriqμoV pou qa diabάzei caraktήreV keiμέnou , μέcri na sunantήseitouV caraktήreV ª.º ή ª!º .O algόriqμoV ektupώnei to plήqoV twn caraktήrwn poudiάbase kaqώV kai to plήqoV eμfanίsewn tou caraktήra ªaº. Oi caraktήreV ª.º kai ª!ºden kataμetroύntai.Qa epilύsouμe thn άskhsh μe th crήsh thV doμήV Όso.AlgόriqμoV Mέtrhsh_CaraktήrwnTέloV Mέtrhsh_Caraktήrwnsunolikό_plήqoV ¬0plήqoV_a ¬0Δiάbase grάμμaΌsogrάμμa ª.º kai grάμμa ª!º epanέlabesunolikό_plήqoV ¬ sunolikό_plήqoV +1

An grάμμa= ªaº tόteplήqoV_a ¬ plήqoV_a+1TέloV_an! Gia thn epόμenh entolήTέloV_epanάlhyhVGrάye sunolikό_plήqoV,plήqoV_aANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.20 EPANA

LHYHNa anaptucqeί algόriqμoV pou qa diabάzei zeύgh ariqμώn . O algόriqμoV qa rwtάei toncrήsth an epiquμeί na sunecίsei thn katacώrhsh nέwn stoiceίwn . Sto tέloV thVepanalhptikήV diadikasίaV qa ektupώnei to plήqoV twn zeugariώn pou diάbase.Se autό to parάdeigμa endeίknutai h crήsh thV doμήV MέcriV_όtou,afoύ epiquμoύμe thn epexergasίa όlwn twn stoiceίwn pou diabάzouμe.AlgόriqμoV Zeύgh_tiμώn


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TέloV Zeύgh_tiμώnplήqoV ¬0Arcή_epanάlhyhVΔiάbase tiμή1,tiμή2plήqoV ¬ plήqoV +1Grάye ªEpiquμeίte na sunecίsete(NAI/OCI);ºΔiάbase apάnthshMέcriV _όtou apάnthsh = ª ºGrάye ªTa zeύgh tiμώn pou diabάsthkan eίnai º,plήqoVANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.21 EPANALHYHNa anaptucqeί algόriqμoV pou qa diabάzei έnan ariqμό (μegalύtero tou 0) kai qaupologίzei to άqroisμa:

=1+2

2+33+44+¼μέcri to na xeperάsei thn tiμή tou ariqμoύ pou diabάsthke. O algόriqμoV qa ektupώneito plήqoV twn όrwn pou creiάsthkan gia ton upologisμό tou .AlgόriqμoV Άqroisμa_ΌrioTέloV Άqroisμa_ΌrioArcή_epanάlhyhVΔiάbase ΌrioMέcriV _όtou

Όrio > 0 ¬1 ¬0Όso


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=12+14+16+18+¼.AlgόriqμoV Άqroisμa_AkrίbeiaTέloV Άqroisμa_AkrίbeiaΌrio ¬0.00001 ¬1

¬0Όso epόμenoV_όroV>Όrio epanέlabe

¬

+ epόμenoV_όroV ¬ +1epόμenoV_όroV ¬1/2epόμenoV_όroV ¬1/(2* )TέloV_epanάlhyhV

Grάye ªOi epanalήyeiV pou apaitήqhkan eίnai º,

ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.23 EPANALHYHNa anaptucqeί algόriqμoV pou qa diabάzei έnan akέraio ariqμό kai qa ektupώnei to

plήqoV twn yhfίwn tou.AV qewrήsouμe gia parάdeigμa ton ariqμό 3275.An diairέsouμe ton ariqμό μe to 10000 .4 yhfίa10^4to akέraio μέroV thV diaίreshV eίnai 0 kai ta yhfίa tou ariqμoύ eίnai 4 όsa kai ta μhde3275 100000,3275Akέraio μέroV Έtsi , gia na upologίsouμe to plήqoV twn yhfίwn arkeί na diairέsouμe ton ariqμό μeto 10 ,to 100,to 1000 klp

μέcri to akέraio μέroV thV diaίreshV na eίnai 0.ΆgnwstoV ariqμόVepanalήyewn.ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.24 EPANA


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LHYHNa anaptucqeί algόriqμoV pou qa diabάzei έnan akέraio ariqμό kai qa ektupώnei toplήqoV twn yhfίwn tou.Δen eίnai gnwstό to plήqoV twn epanalήyewn pou qa creiastoύn , άra qa crhsiμopoiήsouμe Όso.AlgόriqμoV PlήqoV_yhfίwnTέloV PlήqoV_yhfίwnΔiάbase ariqμόVyhfίa ¬1Όso ariqμόV

v10^yhfίa 0 epanέlabeyhfίa ¬ yhfίa+1TέloV_epanάlhyhVGrάye ªTa yhfίa eίnai º,yhfίaALLIΩSAlgόriqμoV PlήqoV_yhfίwnTέloV PlήqoV_yhfίwnΔiάbase ariqμόVyhfίa ¬0Arcή_epanάlhyhVyhfίa ¬ yhfίa+1

ariqμόV ¬ ariqμόV

v10^yhfίaMέcriV_όtouariqμόV=0Grάye ªTa yhfίa eίnai º,yhfίaANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.24 EPANALH

YHΌtan kάpoioV ftarnίzetai ,sunhqίzetai na tou lέne έna tetrayήfio ariqμό .Tόte autόV prosqέtei ta yhfίa tou ariqμoύ ,μέcri an antistoiceί se kάpoio grάμμa thV alfabήtou.Na anaptύxete algόriqμo pou qa diabάzei έnan ariqμό , μe 4 yhfίa ,kai qa entopίzei to grάμμa thV alfabήtou pou antistoiceί sto ftάrnisμa.O algόriqμoV eίnai:AlgόriqμoV Me_tiV_ugeίeV_saVTέloV Me_tiV_ugeίeV_saVArcή_epanάlhyhVΔiάbase ariqμόVMέcriV _όtou (ariqμόV> = 1000) kai (ariqμόV< = 9999) ! 4 yhfίa

Arcή_epanάlhyhVciliάdeV ¬ ariqμόV v 1000! Brίskw tiV ciliάdeVbohqhtikή1 ¬ ariqμόV m 1000 ! Brίskw ton upόloipo ariqμόekatontάdeV ¬ bohqhtikή1 v 100bohqhtikή2 ¬ bohqhtikή1 m 100dekάdeV ¬ bohqhtikή2

v 10μonάdeV ¬ bohqhtikή2 m 10ariqμόV ¬ ciliάdeV+ekatontάdeV+dekάdeV+μonάdeVMέcriV _όtou ariqμόV< = 24


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Grάye ªProkύptei to grάμμa pou brίsketai sth qέsh , ariqμόVMe μia doμή pίnakaantistoicίzouμe ton ariqμό μe togrάμμa.ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.25 EPANALHYH ΈnaV trόpoV upologisμoύ twn trigwnoμetrikώn sunartήsewn stouV upologistέV , eίnai μe toupologisμό twn parakάtw seirώn:hμc=

-

33!+

55!-

77!+¼. kai sunc= 1-

22!+

44!-

66!+¼.Na grafeί algόriqμoV o opoίoV qa diabάzei th gwnίa

se μoίreV kai qa upologίzei to hμίsunhμίtono thV , sύμfwna μe touV parapάnw tύpouV. Poia μporeί na eίnai ta kritήria gia epanalήyewn;Upόdeixh : Na μetatrapeί arcikά h gwnίa

se aktίnia.Upάrcoun dύo trόpoi gia ton terμatisμό thV epanάlhyhV katά ton upologisμό tou aqroίsμat H epanάlhyh terμatίzetai όtan o nέoV όroV tou aqroίsμatoV eίnai μikrόteroV apό μia

tiμή (gia parάdeigμa 0.00001), kai έtsi o όroV autόV den qa tropopoiήsei katά polύ to ά Sugkrίnetai to άqroisμa μe thn pragμatikή tiμή tou hμitόnou/sunhμitόnou thV gwnίaV sunartήseiV glώssaV-ΔeV Tupolόgio )ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.26 EPANA


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LHYH ΈnaV trόpoV upologisμoύ twn trigwnoμetrikώn sunartήsewn stouV upologistέV , eίnai μe toparakάtw seirώn:hμc=

-

33!+

55!-

77!+¼. kai sunc= 1-

22!

+

44!-

66!+¼.AlgόriqμoV TrigwnoμetrίaTέloV Trigwnoμetrίaόrio ¬0.00001p ¬3.14

Δiάbase gwnίa_μoίreVgwnίa ¬ gwnίa_μoίreV*p/180hμίtono ¬ gwnίa

¬1 ¬3Arcή_epanάlhyhVparonoμastήV ¬1

apό 1 Gia μέcri

paronoμastήV ¬ paronoμastήV* TέloV_epanάlhyhVnέoV_όroV¬(-1) ̂ *gwnίa^N/paronoμastήVhμίtono¬ hμίtono+ nέoV_όroV ¬ +1

¬

+2An nέoV_όroV>= 0 tόteApόl_tiμή_nέoV ¬ nέoV_όroVAlliώVApόl_tiμή_nέoV ¬(-1)*nέoV_όroVTέloV_anMέcriV _όtou Apόl_tiμή_nέoV < = όriosunhμίtono ¬1


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¬1

¬2Arcή_epanάlhyhVparonoμastήV ¬1 apό 1 Gia μέcri paronoμastήV ¬ paronoμastήV*

TέloV_epanάlhyhVnέoV_όroV¬(-1)

̂

*gwnίa^N/paronoμastήVsunhμίtono¬ sunhμίtono+ nέoV_όroV

¬

+1 ¬ +2An nέoV_όroV>= 0 tόteApόl_tiμή_nέoV ¬ nέoV_όroVAlliώVApόl_tiμή_nέoV ¬(-1)*nέoV_όroVTέloV_anMέcriV _όtou Apόl_tiμή_nέoV < = όrioGrάye ªTo hμίtono eίnai º,hμίtonoGrάye ªTo sunhμίtono eίnai º,sunhμίtono! Δhμiourgώ paragontikό! Kai na dhμiourgήsw to nέo paragontikό! Metafέroμai na dhμiourgήsw to nέo όroANAPTUXH EFARMOGΩN

∆oμή EpanάlhyhVEP.26 EPANALHYHNa anaptucqeί algόriqμoV pou qa eμfanίzei όlouV touV tέleiouV ariqμoύV sto diάsthμa[2,100]. TέleioV eίnai o akέraioV pou isoύtai μe to άqroisμa twn gnήsiwn diairetώn tou i tέleioi

O PuqagόraV pίsteue όti orisμέnoi ariqμoί, όpwV o 6, prέpei na qewroύntai «tέleioi».TέleioV lέgetai kάqe ariqμόV o opoίoV eίnai ίsoV μe to άqroisμa twn diairetώn tou.ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.27O 6 eίnai TELEIOS diόti eίnai ίsoV μe to άqroisμa twn 1, 2 kai 3kai oi 1,2,3 eίnai oi treiV diairέteV tou (6=1+2+3) .Autό den suμbaίnei oύte μe ton 5, oύte μe ton 7oύte μe ton 8oύte μe kanέna άllo μonoyήfio.Gia na broύμe ton epόμeno tέleio ariqμό creiάzetai scetikήupoμonή diόti epόμenoV tέleioV eίnai o 28 = 1+2+3+4+5+6+7 .Eάn de qelήsouμe na anazhtήsouμe ton epόμeno tέleio ariqμόqa creiasteί pάli μegάlh upoμonή.

Eίnai o ariqμόV 496 = 1+2+3+4+5+6+ . . . +30+31 .Όso gia ton epόμeno, eάn den broύμe άllon trόpo gia thn anazήthsh, aV to afήsouμekalύtera.1+2+3=6EPANAL


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HYHNa anaptucqeί algόriqμoV pou qa eμfanίzei όlouV touV tέleiouV ariqμoύV sto diάsthμa[2,100]. TέleioV eίnai o akέraioV pou isoύtai μe to άqroisμa twn gnήsiwn diairetώn touMia apό tiV idέeV tou Puqagόra ήtan kai όtih teleiόthta scetίzetai μe tiV dunάμeiV tou 2.Paratήrhse dhladή όti όleV oi dunάμeiV tou ariqμoύ 2 apotugcάnounμόliV sto na eίnai tέleioiafoύ to άqroisμa twn diairetώn touVeίnai μikrόtero MONO katά μίa μonάda apό touV ίdiouV .Parάdeigμa:22= 4 ∆iairέteV oi 1,2 Άqroisμa twn diairetώn = 323= 8 ∆iairέteV oi 1,2,4 Άqroisμa twn diairetώn = 724= 16 ∆iairέteV oi 1,2,4,8 Άqroisμa twn diairetώn = 1525= 32 ∆iairέteV oi 1,2,4,8,16 Άqroisμa twn diairetώn = 31

26= 64 ∆iairέteV oi 1,2,4,8,16,32 Άqroisμa twn diairetώn = 63ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.27 EPANALHY

HNa anaptucqeί algόriqμoV pou qa eμfanίzei όlouV touV tέleiouV ariqμoύV sto diάsthμa[2,100]. TέleioV eίnai o akέraioV pou isoύtai μe to άqroisμa twn gnήsiwn diairetώn touO algόriqμoV eίnai:AlgόriqμoV Tέleioi_ariqμoίTέloV Tέleioi_ariqμoί apό 2 Gia μέcri 100άqroisμa ¬0j apό 1 Gia μέcri An m j=0 tόteάqroisμa ¬ άqroisμa+jTέloV_anTέloV_epanάlhyhV

An άqroisμa=2*

tόteGrάye ªO ariqμόV º, , ª eίnai tέleioV ºTέloV_anTέloV_epanάlhyhVAV pάrouμe gia parάdeigμa ton ariqμό =6 , poueίnai tέleioV:j=1 m j 6 m 1 = 0 άqroisμa ¬ άqroisμa+jάqroisμa ¬1


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j=2

m

j 6 m

2 = 0 άqroisμa ¬1+jάqroisμa ¬1+2 =3j=3

m

j 6 m

3 = 0 άqroisμa ¬3+jάqroisμa ¬3+3 =6j=4

m

j 6 m

4 = 0j=5

m

j 6 m

5 = 0j=6

m

j 6 m

6 = 0 άqroisμa ¬6+jάqroisμa ¬6+6 =12άqroisμa = 2* 6 = ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.27 EPANA

LHYHNa anaptucqeί algόriqμoV pou qa eμfanίzei όlouV touV prώtouV ariqμoύV sto diάsthμa[2,100] kaqώV kai to plήqoV touV. PrώtoV eίnai o akέraioV tou opoίou oi μonadikoί diaireίnai h μonάda kai o eautόV tou.ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.28 EPAN

ALHYHp.c. 5(5,1), 7(7,1)oi prώtoi ( m m )O «6» eίnai «ginόμeno» tou «2» kai tou «3», «prokύptei» apό ton 2 kai ton 3.O «30» «prokύptei» apό ton 2, ton 3 kai ton 5,enώ o 17 «den prokύptei» apό kάpoiouV άllouV ariqμoύV.O «17» eίnai PRΩTOS , όpwV kai o 13, o 5, o 7 kai o 11 ,όpwV kai kάqe akέraioV pou den έcei diairέth ektόV fusikά apό ton eautό tou kai apό tonOi PRΩTOI eίnai oi «doμikoί lίqoi» twn (akέraiwn) ariqμώn

kai autό eίnai kάti pou to diέkrinan oi ΈllhneV όtan diapίstwsan όtikάqe ariqμόV μporeί na «gίnei» apό prώtouV ariqμoύV.ΌpwV oi chμikoί agwnίsthkan na prosdiorίsoun ta basikά stoiceίa thV ύlhVkai katέlhxan sta 92 diaforetikά άtoμa,oi ΈllhneV μaqhμatikoί έkanan μia kalή arcή blέpontaV touV PRΩTOUS kάtisan « ATOMA thV ARIQMHTIKHS » san doμikoύV dhladή lίqouV όlwn twn ariqμώn.Na anaptucqeί algόriqμoV pou qa eμfanίzei όlouV touV prώtouV ariqμoύV sto diάsthμa[2,100] kaqώV kai to plήqoV touV. PrώtoV eίnai o akέraioV tou opoίou oi μonadikoί diaireίnai h μonάda kai o eautόV tou.Poioi eίnai oi prώtoi ariqμoί;


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Eύkolh h apάnthsh gia touV «μikroύV» ariqμoύV,dύskolh έwV adύnath gia touV polύ μegάlouV .AV arcίsouμe όμwV apό touV μikroύV.Kat' arcήn kanέnaV prώtoVden μporeί eίnai άrtioV ektόV apό to 2.Sthn periocή twn μonoyήfiwn oi prώtoi eίnai tέsseriV,o 2, o 3, o 5 kai o 7.Sth deύterh dekάda eίnai epίshV tέsseriV,o 11, o 13, o 17, o 19enώ sthn trίth dekάda eίnai dύo, o 23, kai o 29kai sthn tέtarth o 31, kai o 37kai sthn pέμpth o 41, o 43 kai o 47.StouV prώtouV dhladή 50 akέraiouV oi PRΩTOI eίnai dekapέnte ariqμoί.ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.28 EPANALHYH

oi prώtoi (

m

m

)O algόriqμoV eίnai:AlgόriqμoV Prώtoi_ariqμoίTέloV Prώtoi_ariqμoίplήqoV ¬0 apό 2 Gia μέcri 100diairέteV ¬0j apό 1 Gia μέcri An m j=0 tόtediairέteV ¬ diairέteV+1TέloV_anTέloV_epanάlhyhV

An diairέteV=2 tόteGrάye ªO ariqμόV º, , ª eίnai prώtoV ºplήqoV ¬ plήqoV+1TέloV_anTέloV_epanάlhyhVGrάye ªPlήqoV prώtwn :º,plήqoVANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.28 EPAN

ALHYHNa anaptucqeί algόriqμoV pou qa eμfanίzei όlouV touV prώtouV ariqμoύV sto diάsthμa[2,100] kaqώV kai to plήqoV touV. PrώtoV eίnai o akέraioV tou opoίou oi μonadikoί diaireίnai h μonάda kai o eautόV tou.Na anaptucqeί algόriqμoV pou qa diabάzei dύo qetikoύV akέraiouV ariqμoύV kai qaektupώnei to Mέgisto Koinό Δiairέth touV.


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Gia ton upologisμό tou M.K.Δ. dύo akeraίwn μporoύμe na crhsiμopoiήsouμe ton algόriqμo tΔiairoύμe ton μegalύtero dia ton μikrόtero, kai kratoύμe to upόloipo thV diaίreshV .Sth sunέceia diairoύμe ton μikrόtero akέraio dia to upόloipo kai kratάμe to nέo upόloipEpanalaμbάnouμe thn parapάnw diadikasίa ώspou to upόloipo na eίnai 0 kai M.K.Δ. eίnai oANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.29 EPANALHYHp.c. (123,45)=(45,33)=(33,12)=(12,9)=(9,3)=(3,0)Άra M.K.Δ=3Na anaptucqeί algόriqμoV pou qa diabάzei dύo qetikoύV akέraiouV ariqμoύV kai qaektupώnei to Mέgisto Koinό Δiairέth touV.ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.29 EPA

NALHYHO algόriqμoV eίnai:AlgόriqμoV M_K_ΔTέloV M_K_ΔΔiάbase a,bAn a>b tόtediairetέoV ¬ a

diairέthV ¬ bAlliώVdiairetέoV ¬ bdiairέthV ¬ aTέloV_anArcή_epanάlhyhVupόloipo ¬ diairetέoV m diairέthVdiairetέoV ¬ diairέthVdiairέthV ¬ upόloipoMέcriV _όtou upόloipo = 0Grάye ªO MέgistoV KoinόV diairέthV eίnai :º,diairetέoVNa anaptucqeί algόriqμoV pou qa dέcetai έnan akέraio ariqμό kai qa ton analύei se

ginόμeno prώtwn paragόntwn.Ginόμeno prώtwn paragόntwn:ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.30 EPANAL


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HYH60 = 2 * 30μikrόteroV parάgontaV tou 60=2* 2* 15μikrόteroV parάgontaV tou 30=2* 2* 3*μikrόteroV parάgontaV tou 155AlgόriqμoV ParάgonteVTέloV ParάgonteVΔiάbase ariqμόVbohqhtikή ¬ ariqμόV ¬1

¬1Arcή_epanάlhyhV

¬

+1 ¬0Όsobohqhtikήm =0 epanέlabebohqhtikή ¬ bohqhtikή

v

¬ +1TέloV_epanάlhyhV

An

>0 tόteGrάye

,

TέloV_an ¬ * ̂ MέcriV _όtou = ariqμόV8bohqhtikή ¬8 ¬2 ¬08 m 2=0bohqhtikή ¬8 v2=4 ¬1

¬1 ¬11h Epanάlhyh Arcή¼1h Epanάlhyh Όso¼! O ariqμόV diaireίtai forέV μe to .Na anaptucqeί algόriqμoV pou qa dέcetai έnan akέraio ariqμό kai qa ton analύei seginόμeno prώtwn paragόntwn.AlgόriqμoV ParάgonteV

TέloV ParάgonteVΔiάbase ariqμόVbohqhtikή ¬ ariqμόV ¬1 ¬1Arcή_epanάlhyhV

¬

+1 ¬0Όsobohqhtikήm =0 epanέlabe


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bohqhtikή ¬ bohqhtikή v

¬

+1TέloV_epanάlhyhVAn >0 tόteGrάye , TέloV_an ¬ * ̂ MέcriV _όtou = ariqμόV4 m

2=0bohqhtikή ¬4 v2=22h Epanάlhyh Όso¼ ¬2ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.30 EPANALH

YHNa anaptucqeί algόriqμoV pou qa dέcetai έnan akέraio ariqμό kai qa ton analύei seginόμeno prώtwn paragόntwn.AlgόriqμoV ParάgonteVTέloV ParάgonteVΔiάbase ariqμόVbohqhtikή ¬ ariqμόV ¬1 ¬1Arcή_epanάlhyhV ¬ +1 ¬0

Όsobohqhtikήm =0 epanέlabebohqhtikή ¬ bohqhtikή v

¬

+1TέloV_epanάlhyhVAn >0 tόteGrάye , TέloV_an ¬ * ̂ MέcriV _όtou = ariqμόV2 m

2=0bohqhtikή ¬2 v2=1

3h Epanάlhyh Όso¼ ¬3ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.30 EPAN


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ALHYHNa anaptucqeί algόriqμoV pou qa dέcetai έnan akέraio ariqμό kai qa ton analύei seginόμeno prώtwn paragόntwn.AlgόriqμoV ParάgonteVTέloV ParάgonteVΔiάbase ariqμόVbohqhtikή ¬ ariqμόV

¬1 ¬1Arcή_epanάlhyhV ¬ +1

¬0Όsobohqhtikήm

=0 epanέlabebohqhtikή ¬ bohqhtikή v

¬

+1TέloV_epanάlhyhVAn

>0 tόteGrάye , TέloV_an

¬

*

^

MέcriV _όtou = ariqμόV1 m 2=03h Epanάlhyh Όso¼ =3>02,3 ¬1*2^3=8 =8=ariqμόV Tέrμa Epanάlhyh Arcή¼! O ariqμόV diaireίtai3forέV μe to 2.

ANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.30 EPANALHYHO ruqμόV aύxhshV twn kapnistώn sthn Ellάda eίnai 8.5%. An sήμera oi kapnistέVektiμάtai όti ariqμoύn 4.500.000, na anaptucqeί algόriqμoV pou qa ektupώnei to plήqoV

twn kapnistώn se 15 crόnia kaqώV kai to posostό aύxhshV touV s' autό to cronikόdiάsthμa.Iscύei όti ruqμόV aύxhshV= 100* telikή tiμή-arcikή tiμήarcikή tiμήAlgόriqμoV KapnistέVTέloV KapnistέVkapnistέV_sήμera ¬4500000kapnistέV ¬ kapnistέV_sήμera


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ruqμόV ¬8.5/100

apό 1 Gia μέcri 15kapnistέV ¬ kapnistέV+ A_M(kapnistέV*ruqμόV) ! Akέraia tiμήTέloV_epanάlhyhVruqμόV ¬100*(kapnistέV-kapnistέV_sήμera)/kapnistέV_sήμeraGrάye ªKapnistέV qa eίnai º,kapnistέV, ª μe % aύxhsh º,ruqμόVANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.31 EPANALHYHApό έreuneV pou έcei faneί όti μia koinόthta μelissώn upό kanonikέV sunqήkeVanaptύssetai μe ruqμό 4.8% ethsίwV. An έnaV μelissokόμoV diaqέtei μelίssia μe sunolikόpluqhsμό 1200 μέlisseV, se pόsa έth qa xeperάsei th cwrhtikόthta twn kuyelώn toupou eίnai 2000 μέlisseV; Na grafeί algόriqμoV pou qa ektupώnei to zhtoύμeno.Efόson den eίnai gnwstό to plήqoV twn epanalήyewn , qa crhsiμopoiήsouμe th doμήepanάlhyhV MέcriV_όtou kaqώV kai έnan μetrhtή Έth pou qa μetrά tiV epanalήyeiV ,dhladή ta crόnia.

AlgόriqμoV MέlisseVTέloV MέlisseVMέlisseV ¬1200RuqμόV ¬4.8/100 Έth ¬0Όso MέlisseV=6000 epanέlabe! Akέraia tiμήPlhqusμόV ¬PlhqusμόV - A_M(PlhqusμόV *ruqμόV) Έth ¬ Έth+1


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TέloV_epanάlhyhVGrάye ªQa carakthristeί eίdoV proV exafάnish se º,Έth, ª έth ºANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.33 EPANALHYHTo aμfiqέatro tou dήμou diaqέtei 50 kaqίsμata sthn prώth seirά kai se kάqe epόμenhseirά apό tiV sunolikά 15 upάrcei aύxhsh kaqisμάtwn katά 10%. Na anaptucqeίalgόriqμoV pou qa upologίzei kai qa ektupώnei ta kaqίsμata thV teleutaίaV seirάV, kaqώVkai th sunolikή cwrhtikόthta kaqisμάtwn tou aμfiqeάtrou.AlgόriqμoV AμfiqέatroTέloV Aμfiqέatrokaqίsμata ¬50RuqμόV ¬10/100sunolikά_kaqίsμata ¬ kaqίsμata apό 2 Gia μέcri 15 ! GnwstόV ariqμόV epanalήyewnkaqίsμata ¬kaqίsμata + A_M(kaqίsμata *ruqμόV)

sunolikά_kaqίsμata ¬ sunolikά_kaqίsμata +kaqίsμataTέloV_epanάlhyhVGrάye ªTa kaqίsμata thV teleutaίaV seirάV eίnai º,kaqίsμataGrάye ªTo sύnolo twn kaqisμάtwn eίnai º,sunolikά_kaqίsμataANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.34 EPANALH

YH! Δeύterh seirά kai exήV.O dήμoV apofάsise na topoqetήsei nέa kaqίsμata sto aμfiqέatro. Sthn prώth seirά topoqetkaqίsμata enώ se kάqe epόμenh seirά prostίqentai 6 kaqίsμata. To kόstoV kάqe kaqίsμatoVenώ ta diaqέsiμa crήμata( ) eίnai 20000 €. Na anaptucqeί algόriqμoV pou qa ektupώnariqμό kaqisμάtwn pou μporoύn na topoqethqoύn , kaqώV kai to crhμatikό posό pou perisseMia lύsh qa ήtan na crhsiμopoiήsouμe th doμή Όso wV exήV:Όso sunolikό_kόstoV


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ANALHYHO dήμoV apofάsise na topoqetήsei nέa kaqίsμata sto aμfiqέatro. Sthn prώth seirά topoqetkaqίsμata enώ se kάqe epόμenh seirά prostίqentai 6 kaqίsμata. To kόstoV kάqe kaqίsμatoVenώ ta diaqέsiμa crήμata(

) eίnai 20000 €. Na anaptucqeί algόriqμoV pou qa ektupώnariqμό kaqisμάtwn pou μporoύn na topoqethqoύn , kaqώV kai to crhμatikό posό pou perisseAlgόriqμoV Aμfiqέatro2TέloV Aμfiqέatro2

¬20000kaqίsμata ¬50sun_kόstoV ¬0trέcon_kόstoV ¬ kaqίsμata* 40! kόstoV 1hV seirάVsun_kaqίsμata ¬0Όso sun_kόstoV+trέcon_kόstoV


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O dήμoV apofάsise na topoqetήsei nέa kaqίsμata sto aμfiqέatro. Sthn prώth seirά topoqetkaqίsμata enώ se kάqe epόμenh seirά prostίqentai 6 kaqίsμata. To kόstoV kάqe kaqίsμatoVenώ ta diaqέsiμa crήμata( ) eίnai 20000 €. Na anaptucqeί algόriqμoV pou qa ektupώnariqμό kaqisμάtwn pou μporoύn na topoqethqoύn , kaqώV kai to crhμatikό posό pou perisseAlgόriqμoV Aμfiqέatro2TέloV Aμfiqέatro2 ¬20000kaqίsμata ¬50sun_kόstoV ¬0trέcon_kόstoV ¬ kaqίsμata* 40! kόstoV 1hV seirάVsun_kaqίsμata ¬0έxodoV¬ YeudήVArcή_epanάlhyhVAn sun_kόstoV+trέcon_kόstoV


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TέloV_anTέloV_epanάlhyhVposostό ¬((plήqoV-arcikό_plήqoV)/arcikό_plήqoV)*100Grάye ªTelikόV plhqusμόV: º,plήqoVGrάye ªPosostό aύxhsh: º,posostόANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.36 EPANALHYHO μisqόV tou kύriou Papadόpoulou eίnai 1250 €, enώ sύμfwna μe to μisqolόgio auxάnetaikatά 11% ethsίwV. Kάqe μήna έcei apofasίsei na apotaμieύsei to 9% tou μisqoύ gia toόneiro tou , pou eίnai h agorά fouskwtoύ skάfouV. Na anaptucqeί algόriqμόV pou qaupologίzei kai qa ektupώnei se pόsouV μήneV qa katorqώsei na sugkentrώsei toapaitoύμeno posό, ώste na agorάsei fouskwtό axίaV 7000 €.AlgόriqμoV FouskwtόTέloV FouskwtόμisqόV ¬1250

sugkentrwqέn_posό ¬0μήneV ¬0Όso sugkentrwqέn_posό< 7000 epanέlabesugkentrwqέn_posό ¬sugkentrwqέn_posό + 0.09*μisqόVμήneV ¬ μήneV+1An μήneV m 12=0 tόteμisqόV ¬ μisqόV+(11/100)*μisqόVTέloV_epanάlhyhVGrάye ªTo posό qa sugkentrwqeί se : º,μήneV,ªμήneVºTέloV_anANAPTUXH EFARMOGΩN

∆oμή EpanάlhyhVEP.37 EPANALHYHMe thn ekkίnhsh thV suskeuήV enόV kinhtoύ zhteίtai o kwdikόV prόsbashV

kai ocrήsthV έcei treiV eukairίeV gia thn eisagwgή tou. Na anaptucqeί o algόriqμoV pou ektelto kinhtό :zhtάei 3 forέV ton kwdikό prόsbashV (an den έcei eisacqeί swstά) kai sthn

perίptwsh triplήV apotucίaV ektupώnei to μήnuμa ªH kάrta

kleidώqhke. Parakalώeisάgete ton kwdikό

ºAlgόriqμoV Eisagwgή_RINTέloV Eisagwgή_RINΔedoμέna //

//apotucίeV ¬0Δiάbase kwdikόVΌso(kwdikόV RIN) kai (apotucίeV3) epanέlabeapotucίeV ¬ apotucίeV +1


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Grάye ªLάqoV kwdikόV. UpόloipeV dokiμέV:º 3-apotucίeV, ªΔokiμάste xanά . ºΔiάbase kwdikόVTέloV_epanάlhyhVAn kwdikόV= RIN tόteGrάye ªKalώV ήrqate sto dίktuo saV¼.. ºAlliώVTέloV_anGrάye ªH kάrta kleidώqhke. Parakalώ eisάgete ton kwdikό

. ºANAPTUXH EFARMOGΩN∆oμή EpanάlhyhVEP.38 EPANALHYHANAPTUXH EFARMOGΩN

56385272 Δομή Επανάληψης Ασκήσεις Επανάληψης Πριν Τις...

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