4MaThe Binomth-ial Theorem-r1 - The Binomial Theorem The Binomial Theorem Theorem Given any numbers a and b and any nonnegative integer n n n i i a

4MaThe Binomth-ial Theorem-r1 - The Binomial Theorem...

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The Binomial Theorem
The Binomial Theorem Theorem: Given any numbers a and b and any nonnegative integer n , n i i i n n b a i n b a 0 .
The Binomial Theorem Proof: Use induction on n . Base case: Let n = 0. Then ( a + b ) 0 = 1 and Therefore, the statement is true when n = 0. 0 0 0 0 0 0 . 1 0 0 0 i i i b a b a i
Proof, continued Inductive step Suppose the statement is true when n = k for some k 0. Then  k i i i k k i i i k k i i i k k k b a i k b a i k b a i k b a b a b a b a 0 1 0 1 0 1
Proof, continued k i k i i k k k i k i k i i k i i k k k i k i k i i k i i k k b b a i k i k a b b a i k b a i k a b b a i k b a i k a 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1
Proof, continued

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