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S@!] eYt =󋰋R64Cґ 3!J u ,Q޽^4 #||"|9B,JB!i+(0^7M=RppPT$B`!^o6k`: t1؎ IpWh\q0v9/IvU, aP| qYg6uU{BQ7./g_HZeﶌ>13nve:I.RX9A\Nru(׌I^bܭ"&[5-p!a5y69+ J^5f",պ|K[x2$ >NwSv꠾q2$]i³|;w7㯍D$2$.杫*8KL 0AA f@8`ʚ;ʚ;g4HdHd n 08ppp@ <4dddd8A10l. e0___PPT10 pp%___PPT9nO9@[?VyPNG  IHDRysRGBPLTEZZZEfP cmPPJCmp07128I tRNS ATIDAT Av0d4 U TCX*XSŸJFeπ t*hS u5CB>_IENDB`?  %`!Geometrijski niz   Uvodni deo - primeriNekada davno u Persiji je ~iveo kralj iju je ker miljenicu oteo razbojnik. Kralju nikako nije uspevalo da spase ker dok se nije pojavio, naravno, siromaani selja ki sin. Kralju je kao nagradu za spasavanje keri zatra~io onoliko paenice koliko staje na aahovsku tablu i to na sledei na in: na prvo polje jedno zrno, na drugo polje dva zrna, na tree etiri, na etvrto osam itd. Kralj je, ne znajui za geometrijski niz, lakomisleno pristao na pogodbu mislei kako je jeftino proaao. Meutim, kada je ker bila spaaena i kada je siromah doaao po svoju paenicu, ispostavilo se da ne bi bilo dovoljno ni nekoliko godianjih ~etvi u celoj Persiji da isplati spasioca svoje ljubimice. Naime, ukupna koli ina paenice je bila jednaka zbiru geometrijskog niza od lana koli nika sa po etnim lanom , tj. ukupan broj zrna paenice bi bio:2AP3@ Uvodni deo - primeri|Zamislite da Ahil tr i protiv kornja e. Ahil tr i 10 puta br~e, ali po inje od ta ke A, 100 metara iza kornja e koja je u ta ki K1 (kornja i, koja je sporija je data prednost). Da bi prestigao kornja u, Ahil mora prvo doi do ta ke K1. Meutim, kada je Ahil stigao do ta ke K1, kornja a je preala 10 metara i sad je u ta ki K2. Ponovo Ahil tr i do K2. Ali, kao i ranije, kornja a je metar ispred njega u ta ki K3. Koliko god nastavili tu trku dalje, kornja a e uvek imati prednost nad Ahilom ( ma koliko mala ona bila ) i Ahil nikada nee stii kornja u! Ovo je jedan od tzv. Zenonovih paradoksa o kretanju koje je opisao joa Aristotel. U stvari se opet radi o geometrijskom nizu, doduae beskona nom.2P Uvodni deo - primeri`ta zna i da supstanca ima vreme poluraspada od n godina? To zna i da svakih n godina polovina njene koli ine nestaje. Posle prvih n godina, polovina koli ine ostaje, nakon sledeih n polovina od te polovine, dakle etvrtina ukupne koli ine ostaje. Zatim posle sledeih n godina polovina te koli ine, odnosno osmina po etne i tako dalje. Dakle, posle mn godina, ukoliko je po etna koli ina supstance x, koli ina koja ostaje je (0,5)mx, ato je opet primena geometrijskog niza. 4. *Uvodni deo - primeriUlaganjem odreene koli ine para G u banku sa kamatnom stopom p, nakon prve godine svota u banci iznosi: G ( 1 + p ) , nakon druge: G ( 1 + p ) ( 1 + p ), a nakon n-te: G ( 1 + p )n Dakle, opet geometrijski niz l_{ :  np   +8Glavni deo  definicija nizaOpati lan niza (Glavni deo  primeri |Nakon zavraene osnovne akole, kao nagradu za Vukovu diplomu, stric je Milanu oro io 1000 evra na 4 godine u banci gde je kamatna stopa 6 %. Koliko e para imati Milan kada krene na fakultet?4Aa5A:4 8Zavrani deo  domai zadatakU odeljenju od 32 u enika, samo jedan u enik je na pismenom iz matematike reaio peti zadatak. Nakon 10 minuta od po etka asa, on je u roku od 5 minuta reaen zadatak poslao dvojici u enika. Svaki od te dvojice je u roku od 5 minuta poslao po dvojici u enika isti zadatak. Oni su u narednih 5 minuta uradili isto. 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