unutra-12330.pdf - 1 Nizovi i redovi 1 2 3 4 5 6 7 Pojam niza 2 Aritmetiˇcki niz 3 Geometrijski niz.

unutra-12330.pdf - 1 Nizovi i redovi 1 2 3 4 5 6 7 Pojam...

This preview shows page 1 - 4 out of 16 pages.

1 Nizovi iredovi 1. Pojam niza . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Aritmetiˇcki niz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3. Geometrijski niz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4. Konvergentni nizovi . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5. Geometrijski red . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 6. Potroˇsaˇcki kredit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 7. Sloˇzeni kamatni raˇcun . . . . . . . . . . . . . . . . . . . . . . . . 27
Image of page 1

Subscribe to view the full document.

NIZOVI I REDOVI 1.1. Pojam niza U razliˇcitim ˇcasopisima za enigmatiku i testovima inteligencije nailazimo na zadatke tipa: Nastavi niz: 1, 4, 9, 16, 25, . . . . U takvim se zadacima oˇcekuje da ˇcitatelj na temelju danih elemenata uoˇci neko pra- vilo, te da na temelju njega napiˇse jedan ili viˇse ˇclanova niza koji nedostaju. No, ˇsto je niz? U prethodnom nizu prvi ˇclan je broj 1, drugi ˇclan je broj 4, tre´ci je broj 9. Dakle, svakom prirodnom broju pridruˇzen je neki broj, tj. imamo pridruˇzivanje 1 1 4 16 2 4 5 25 3 9 n n 2 Ovo razmatranje vrijedi i op´cenito. Niz realnih brojeva je funkcija a koja svakom prirodnom broju n pridruˇzuje neki realni broj a n . Kra´ce, niz a je funkcija a : N R , n a n . Broj a n zove se op´ci ˇclan niza. Za niz osim oznake a ˇcesto koristimo i oznaku ( a n ) . Niz moˇzemo zadati na razne naˇcine:formulom, rekurzijom, opisno, grafiˇcki.Mi´cemo ih zadavati formulom, tj. za op´ci ˇclan nizaanbit ´ce dana formula po kojojse taj niz raˇcuna. Tako, na primjer, formula za op´ci ˇclan niza s poˇcetka teksta glasi a n = n 2 . Primjer 1. a)Napiˇsimo nekoliko prvih ˇclanova niza parnih brojeva i formulu za op´ci ˇclan tog niza.b)Napiˇsimo prvih pet ˇclanova niza zadanog formuloman=n+12n. a) Niz parnih brojeva poˇcinje ovako 2, 4, 6, 8, 10, 12, . . . . Formula koja opisuje op´ci ˇclan tog niza je a n = 2 n . b) a 1 = 1 + 1 2 · 1 = 1, a 2 = 2 + 1 2 · 2 = 3 4 , a 3 = 3 + 1 2 · 3 = 4 6 = 2 3 , a 4 = 4 + 1 2 · 4 = 5 8 , a 5 = 5 + 1 2 · 5 = 6 10 = 3 5 . Ako promatramo samo nekoliko prvih ˇclanova niza tada govorimo o konaˇcnom nizu: a 1 , a 2 , a 3 , . . . , a k . 2
Image of page 2
1.2. ARITMETI ˇ CKI NIZ Zadaci 1.1. 1. Napiˇsi prva 4 ˇclana niza zadanog formulom: a) a n = 2 n + 1 ; b) a n = n 2 2 ; c) a n = 2 n ; d) a n = 2 n 3 1 . 2. Niz je zadan formulom a n = n 2 1 n . Izraˇcunaj a n , ako je n jednako: a) 4; b) 10; c) 11; d) 20. 3.Napiˇsi formulu za op´ci ˇclan niza kojemu je dano prvih pet ˇclanova: 1.2. Aritmetiˇcki niz U ovom poglavlju detaljnije ´cemo prouˇciti dvije vrste nizova koji imaju primjenu u financijskoj matematici, a i drugdje. To su aritmetiˇcki i geometrijski niz. Niz ( a n ) zovemo aritmetiˇckim nizom ako je razlika uzastopnih ˇclanova niza isti broj d , tj. ako je a n + 1 a n = d za svaki n N . Na primjer, niz neparnih brojeva 1, 3, 5, 7, 9, . . . je jedan aritmetiˇcki niz, jer je razlika uzastopnih ˇclanova jednaka d = 2. Broj d zovemo razlika ili diferencija aritmetiˇckog niza. Za drugi ˇclan niza vrijedi: a 2 a 1 = d , tj. a 2 = a 1 + d . Za tre´ci ˇclan niza vrijedi: a 3 a 2 = d , a 3 = a 2 + d =( a 1 + d )+ d = a 1 + 2 d .
Image of page 3

Subscribe to view the full document.

Image of page 4
  • Fall '19
  • David J. Malan

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask You can ask ( soon) You can ask (will expire )
Answers in as fast as 15 minutes