MATEMATIKA 1 Senka Banic PREDAVANJA (grupa G1): utorak i ...

Teorem 4.1. (Binomni poucak)Za svaki n 2 N vrijedi(x + y) n =nXk=0 nkx n k y k :Dokaz. Matematickom indukcijom.Raspis:nPnkxn k y k = n 0xn 0 y 0 + n 1xn 1 y 1 + :::+k=0+ nn 1xn (n 1) y n 1 + n nxn n y n == x n + nx n 1 y + ::: + nxy n 1 + y nPrimjer:(x + y) 4 = x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + y 4(x + y) 5 = x 5 + 5x 4 y + 10x 3 y 2 + 10x 2 y 3 + 5xy 4 + y 5Domaći rad: (x + y) 6 =? (x y) 6 =? (x y) n = ? P n ( 1) k n kxn k y k = x n nx n 1 y + k=0 xn 2 y 2 n xn 3 y 3 + :::: + ( 1) n y nn23