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Binomial Theorem

Consider the power of a binomial where is a natural number. The binomial expansion can be written as a sum of terms.
binomial theorem expansion - combinations

Each of the terms are the combinations of objects taken at a time. The values of are the same as those in the row of Pascal's triangle. Therefore, the binomial expansion can also be written in terms of the Pascal's Triangle's values of the row.

binomial theorem expansion - pascal triangle


To prove this theorem, a method called Mathematical Induction will be used. First the small cases need to be examined. Consider that
Since and are both equal to the theorem holds for this case. Now consider
On this case, and are equal and is equal to making this case also true. Following this, it can be assumed that the theorem holds for a natural number
binomial theorem expansion - combinations with k
To prove the theorem, it needs to be shown that the theorem holds for the value of This can be done by multiplying the case above by
To expand the binomial, multiply by the binomial expansion of This has to be done carefully, as distributing and modify the terms on the binomial expansion.
binomial expansion - big
Examining the resulting terms closely, it can be noted that there are terms that can be factored together. For example, consider the following.
Using these examples, it is possible to find a way to write these terms. Let be a natural number from to Using this variable, factored terms can be written as follows.
Now the sum between parenthesis will be examined to see if it can be simplified. The expressions for the combinations will be used to do so.

Considering this result, the repeating terms can be rewritten to simplify the binomial expansion.
binomial expansion - compact

Looking at the binomial expansion, it can be noted that the repeated terms are written like the theorem. The remaining terms and can be rewritten considering the values of and ·

Combination Formula Simplify

Since of the values on the table are equal to these can be interchanged to rewrite the expression one last time.

binomial expansion - final

Therefore, the result holds for in the case that which finish the proof.