Binomial theorem: Difference between revisions

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In [[elementary algebra]], the '''binomial theorem''' is the identity that states that for any non-negative [[integer]] ''n'',
In [[elementary algebra]], the '''binomial theorem''' or the binomial expansion is the identity that states that for any non-negative [[integer]] ''n'',


: <math> (x + y)^n = \sum_{k=0}^n {n \choose k} x^k y^{n-k}, </math>
: <math> (x + y)^n = \sum_{k=0}^n {n \choose k} x^k y^{n-k}, </math>

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In elementary algebra, the binomial theorem or the binomial expansion is the identity that states that for any non-negative integer n,

or, equivalently,

where

is a binomial coefficient.

One way to prove this identity is by mathematical induction.

Proof:

Base case: n = 0

Induction case: Now suppose that it is true for n : and prove it for n + 1.

and the proof is complete.

The first several cases

Newton's binomial theorem

There is also Newton's binomial theorem, proved by Isaac Newton, that goes beyond elementary algebra into mathematical analysis, which expands the same sum (x + y)n as an infinite series when n is not an integer or is not positive.