The Binomial Theorem States That for Any Real Numbers a and B

question 6

Essay

The binomial theorem states that for any real numbers a and b, $$( a + b ) ^ { n } = \sum _ { k = 0 } ^ { n } \left( \begin{array} { l } n \\k\end{array} \right) a ^ { n - k } b ^ { k } \quad \text { for any integer } n \geq 0$$
Use this theorem to show that for any integer $$n \geq 0 , \sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } n \\k\end{array} \right) 3 ^ { n - k } 2 ^ { k } = 1$$

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Definitions:

Capital

Economic resources that are used by firms to produce goods or services, including buildings, machinery, and equipment.

Capitalist Country

A country that operates on the economic system of capitalism, where private individuals own the means of production and operate for profit.

Resources

The assets, materials, and inputs used to produce goods and services, including natural resources, human talent, and capital.

Fascism

A political ideology characterized by authoritarian power, dictatorial leadership, and the suppression of democratic institutions and societal opposition.