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In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, the power (+) expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying + = and the coefficient of each term is a specific positive integer ...
In 370 BC, Plato's Parmenides may have contained traces of an early example of an implicit inductive proof, [5] however, the earliest implicit proof by mathematical induction was written by al-Karaji around 1000 AD, who applied it to arithmetic sequences to prove the binomial theorem and properties of Pascal's triangle.
The proof of the general Leibniz rule [2]: 68–69 proceeds by induction. Let f {\displaystyle f} and g {\displaystyle g} be n {\displaystyle n} -times differentiable functions. The base case when n = 1 {\displaystyle n=1} claims that: ( f g ) ′ = f ′ g + f g ′ , {\displaystyle (fg)'=f'g+fg',} which is the usual product rule and is known ...
2.2 Inductive and algebraic proofs. 2.2.1 Inductive proof. ... by the binomial theorem, ... This identity can be proven by mathematical induction on ...
This proof of the multinomial theorem uses the binomial theorem and induction on m.. First, for m = 1, both sides equal x 1 n since there is only one term k 1 = n in the sum. For the induction step, suppose the multinomial theorem holds for m.
Using the induction hypothesis, we have that k p ≡ k (mod p); and, trivially, 1 p = 1. Thus (+) + (), which is the statement of the theorem for a = k+1. ∎ In order to prove the lemma, we must introduce the binomial theorem, which states that for any positive integer n,
Among the applications of the product rule is a proof that = when n is a positive integer (this rule is true even if n is not positive or is not an integer, but the proof of that must rely on other methods). The proof is by mathematical induction on the exponent n.
where the above convention for the coefficients of the polynomials agrees with the definition of the binomial coefficients, because both give zero for all i > m and j > n, respectively. By comparing coefficients of x r , Vandermonde's identity follows for all integers r with 0 ≤ r ≤ m + n .