Use the Binomial Theorem to Prove the Following

question 57

Short Answer

Use the binomial theorem to prove the following: $$\left( \begin{array} { c } 100 \\0\end{array} \right) + \left( \begin{array} { c } 100 \\2\end{array} \right) + \left( \begin{array} { c } 100 \\4\end{array} \right) + \left( \begin{array} { c } 100 \\6\end{array} \right) + \cdots + \left( \begin{array} { c } 100 \\98\end{array} \right) + \left( \begin{array} { c } 100 \\100\end{array} \right) = \left( \begin{array} { c } 100 \\1\end{array} \right) + \left( \begin{array} { c } 100 \\3\end{array} \right) + \left( \begin{array} { c } 100 \\5\end{array} \right) + \cdots + \left( \begin{array} { c } 100 \\97\end{array} \right) + \left( \begin{array} { c } 100 \\99\end{array} \right)$$

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Definitions:

Universal Instantiation

A logical rule that allows for the deduction of a particular instance of a statement from a universal statement.

Existential Generalization

A logical operation that infers the existence of at least one individual satisfying a certain property from a specific instance.

Quantifier Negation

The process of applying negation to a quantified statement, potentially altering its meaning significantly.

Universal Generalization

A form of logical inference where a general statement is made about all members of a set based on sufficient evidence or instances.