BinomialTheorem

Binomial theorem (English)

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( x + y )ⁿ

n
∑
i=0




n
i




xⁱ y^{( n−i )}

xⁿ + nx^{( n−1 )} y +




n
2




x^{( n−2 )} y²

+ ... + nxy^{( n −1)} + yⁿ

For any natural number n and complex numbers x,y we have

(x+y)ⁿ =

n
∑
r=0




n
r




x^ry^n−r.

This may be proved by induction, using the result




n+1
r







n
r







n
r−1




the formula which forms the basis of Pascal's triangle.