Nejednakosti s faktorijelima

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1 Osječki matematički list 7007, Nejedakosti s faktorijelima Ilija Ilišević Sažetak Opisae su tehike kako se mogu dokazati ejedakosti koje sadrže faktorijele Spomeute tehike su ilustrirae a izu zaimljivih zadataka koji su prilagod ei učeicima sredjih škola Ključe riječi: faktorijeli, ejedakosti Iequalities with Factorials Abstract The methods for provig iequalities with factorials are described Those methods are demostrated o a umber of iterestig exercises which have bee adapted to high school pupils Key words: factorial, iequalities Umožak svih prirodih brojeva od do ozačava se s! i čita faktorijel Dakle,! 3 Takod er se defiira da je 0! i! Umožak prvih parih prirodih brojeva ozačava se sa!! i čita dvostruki faktorijel, tj!! 4 6, a umožak prvih eparih prirodih brojeva se ozačava sa!! i čita dvostruki faktorijel, dakle!! Takod er se defiira da je 0!! i!! Jaso je da je!! 3!!! te da je!! 3!!! Riješimo sada ekoliko zadataka o ejedakostima s faktorijelima Pritom am je potrebo, poglavito, pozavaje aritmetičko-geometrijske ejedakosti i metode matematičke idukcije III gimazija, Kamila Firigera 4, HR-3000 Osijek

2 8 Ilija Ilišević Zadatak Dokažite da vrijedi ejedakost! ! 008! < Rješeje! ! 008!! 3 +! 3! 3! +!! 3! ! 008! 007! 008! 008! < Zadatak Dokažite da za svaki prirodi broj veći od vrijedi ejedakost Rješeje Kako je k 3 5 k k k + k k >! k k k za k, pri čemu jedakost vrijedi ako i samo ako je k, to je k za k vrijedi stroga ejedakost Uvrštavajem u ovu ejedakost k,,, dobivamo, redom, >, k 0,,, a odatle 3 >! Zadatak 3 Dokažite da za svaki prirodi broj vrijedi ejedakost! < + Rješeje Daa ejedakost ekvivaleta je sa! <+, tj sa! < +

3 Nejedakosti s faktorijelima 83 Dokažimo ovu ejedakost Kako je prema aritmetičko-geometrijskoj AG ejedakosti ++ + >, to je + > +!, pa je >! Zadatak 4 Dokažite da za svaki prirodi broj vrijedi ejedakost ! Rješeje Prema AG-ejedakosti je! Jedakost vrijedi ako i samo ako je Zadatak 5 Dokažite da za svaki prirodi broj vrijedi ejedakost! Rješeje Lijevu strau ejedakosti apišimo u obliku! Sada dokažimo da za k N i k vrijedi k k : k k k kk 0 k kk 0 kk 0 Kako je posljedja ejedakost toča, to je k k Ako u ovu ejedakost uvrstimo k,, 3,,dobivamo, redom,,, 3,,, a odatle! Zadatak 6 Dokažite da za svaki prirodi broj vrijedi ejedakost!

4 84 Ilija Ilišević Rješeje Daa ejedakost ekvivaleta je sa!, tj sa! Dokažimo posljedju ejedakost Za i vrijedi jedakost Za svaki prirodi broj 3 je, prema AG-ejedakosti, <! Zadatak 7 Dokažite da za svaki prirodi broj vrijedi ejedakost! Rješeje Nejedakost ćemo dokazati metodom matematičke idukcije Za ejedakost vrijedi jer je Pretpostavimo da tvrdja vrijedi za eki prirodi broj i dokažimo da vrijedi i za sljedeći prirodi broj +, tj da je +! Kako je +, to je +!! + Prema pricipu matematičke idukcije tvrdja vrijedi za svaki prirodi broj Zadatak 8 Dokažite da za svaki prirodi broj 6 vrijedi ejedakost! < Rješeje Za 6 tvrdja vrijedi jer je 6! 70 < Pretpostavimo da tvrdja vrijedi za eki prirodi broj 6 i dokažimo da vrijedi i za sljedeći prirodi broj + tj da je Nejedakost ćemo dokazati ako dokažemo da vrijedi + + +! < <, jer iz iduktive pretpostavke! < i ejedakosti slijedi

5 Nejedakosti s faktorijelima 85 Pretpostavimo da ejedakost ije toča, tj da za eki prirodi broj 6 vrijedi Imamo Kako posljedja ejedakost e vrijedi i za jeda prirodi broj 6, to je dokazao, a s tim i Zadatak 9 Dokažite da za svaki priroda broj vrijedi ejedakost 4 + <!! Rješeje Za tvrdja vrijedi jer je ! 3 < 6! Pretpostavimo da tvrdja vrijedi za eki priroda broj i dokažimo da vrijedi i za sljedeći priroda broj + tj da je 4 + +! + < +! Imamo <! +! + +! +! + + +!! +5 + < +!! ! +! +!! + Zadatak 0 Dokažite da za sve eegative cijele brojeve p i q vrijedi ejedakost p! q! q q! p + q! Rješeje Nejedakost ćemo dokazati matematičkom idukcijom po q Za q 0 tvrdja vrijedi jer je p! p!

6 86 Ilija Ilišević Pretpostavimo sada da tvrdja vrijedi za eki q N 0 i dokažimo da vrijedi i za q +, tj da je Kako je p! q +! q+ q +! p + q +! p! q +!p! q! q + q + q q! p + q! q + q + q q! p + q! q +p + q + q+ q! p + q! p + q + q + q+ q +! p + q +!, to ejedakost vrijedi i za q + i za proizvolja p N 0 Prema tome, dokazali smo da tvrdja vrijedi za bilo koje brojeve p, q N 0 Zadatak Dokažite da za svaki priroda broj vrijedi ejedakost!!!! < + Rješeje Ozačimo li lijevu strau ejedakosti sa x, imamo x!!!! , pa je < x Dakle, x< + Zadaci za vježbu Dokažite da za svaki priroda broj vrijedi ejedakost! +! + 3! + +! < Dokažite da za svaki priroda broj 4 vrijedi ejedakost!

7 Nejedakosti s faktorijelima 87 3 Dokažite da za svaki priroda broj 3 vrijedi ejedakost! < 4 Dokažite da za svaki priroda broj vrijedi ejedakost! 5 Dokažite da za svaki priroda broj > vrijedi ejedakost! 4!! > +! 6 Dokažite da za svaki priroda broj vrijedi ejedakost <! 3 7 Dokažite da za svaki priroda broj vrijedi ejedakost 3! > 3 Literatura [] I Bušek, Rešeé maturití úlohy z matematiky, Stati pedagogicke akladatelstvi, Praha, 985 [] B Stojaović, Zbirka zadataka iz matematike, Svjetlost, Sarajevo, 985 [3] M P Vasić, R R Jaić, O Mitriović, d D Tošić, Matematički priručik za takmičeja sredjoškolaca i prijeme ispite a fakultetima, Grad eviska kjiga, Beograd, 987 [4] M Željko, Rešee aloge iz matematike z državih i izbirih tekmovaj - IVdel, Društvo matematikov, fizikov i astroomov Sloveije, Ljubljaa, 996

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