Proof of the Binomial Theorem 12.3 - UCSD …

Proof of the Binomial Theorem Theoremsays that: For all real numbersaandband non-negative integersn,(a+b)n=n r=0(nr)arbn example,(a+b)0= 1,(a+b)1=a+b,(a+b)2=a2+ 2ab+b2,(a+b)3=a3+ 3a2b+ 3ab2+ (n) be the statement that for all real numbersaandb, (a+b)n= nr=0(nr)arbn Base Case is easy to we prove the Inductive thatk Z is such that inductive hypothesis (the formula forn=k, , the statementP(k))(a+b)k=k r=0(kr)arbk r.(1)We want to prove that inductive conclusion (the formula forn=k+ 1, , the statementP(k+ 1))(a+b)k+1=k+1 r=0(k+ 1r)arbk+1 r.(2)We compute that(a+b)k+1= (a+b)k (a+b)(3)=k r=0(kr)arbk r (a+b) by the inductive hypothesis=k r=0(kr)ar+1bk r+k r=0(kr)arbk+1 rby the distributive property;indeed, when we multiplyarbk rin line 2 bya,the power ofaincreases by 1 to getar+1bk rin thefirst term in line 3.

Proof of the Binomial Theorem 12.3.1 The Binomial Theorem says that: For all real numbers a and b and non-negative integers n, (a+ b)n = Xn r=0 n r arbn r: For example,

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  Theorem, Binomial theorem, Binomial

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