BINOMNI KOEFICIJENTI
BINOMNI KOEFICIJENTI
BINOMNI KOEFICIJENTI
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<strong>BINOMNI</strong> TEOREM<br />
Dokaz I:<br />
( )<br />
n∑<br />
(x + y) n+1 = (x + y)(x + y) n n<br />
= (x + y) x n−k y k<br />
k<br />
k=0<br />
( n∑<br />
( ) ( n∑<br />
( )<br />
n<br />
= x<br />
)x n−k y k n<br />
+ y<br />
)x n−k y k<br />
k<br />
k<br />
k=0 k=0<br />
( ) ( )<br />
( ) ( )<br />
n<br />
n∑<br />
= x n+1 n ∑n−1<br />
+ x n+1−k y k n<br />
+ x n−k y k+1 n<br />
+ y n+1<br />
0 k<br />
k<br />
n<br />
k=1<br />
k=0<br />
( ) ( )<br />
( ) ( )<br />
n<br />
n∑<br />
= x n+1 n<br />
n∑<br />
+ x n+1−k y k n<br />
+ x n+1−k y k n<br />
+<br />
0 k<br />
k − 1<br />
n<br />
k=1<br />
k=1<br />
[( ) ( )]<br />
n∑<br />
= x n+1 n n<br />
+ + x n+1−k y k + y n+1<br />
k k − 1<br />
= x n+1 +<br />
k=1<br />
(<br />
n∑<br />
k=1<br />
Formula je točna za n + 1.<br />
n + 1<br />
k<br />
)<br />
∑n+1<br />
x n+1−k y k + y n+1 = x n+1−k y k .<br />
k=0<br />
y n+1<br />
() 21. studenog 2011. 16 / 27