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:

Memory Engram

A hypothetical unit or physical substrate within the brain theorized to store memories, encompassing the changes in neural networks that represent a single memory.

Long-Term Memory

The phase of memory responsible for storing information for extended periods, ranging from days to a lifetime.

Declarative Memory

The type of long-term memory that involves the capacity to recall factual information, experiences, and concepts.

Cerebral Cortex

The outer layer of the cerebrum in the brain, involved in high-level functions such as thought, memory, and consciousness.