A Sequence $c _ { 0 } , c _ { 1 } , c _ { 2 } , \ldots$

question 10

Essay

A sequence $c _ { 0 } , c _ { 1 } , c _ { 2 } , \ldots$ satisfies the recurrence relation $c _ { k } = 6 c _ { k - 1 } - 9 c _ { k - 2 }$ with initial conditions $c _ { 0 } = 1$ and $c _ { 1 } = 6$ . Find an explicit formula for the sequence.

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Definitions:

Environmental Pressures

Forces or conditions within an environment that influence the survival, behavior, and evolution of organisms or groups.

Cultural Adaptation

involves adjusting and acclimatizing to a new culture, incorporating its practices and values while often maintaining aspects of one's original culture.

Equilibrium

A state of balance or stability within a system, where various forces or influences are in equal proportion.

Residential School System

A network of boarding schools for Indigenous peoples, primarily in Canada and the United States, intended to assimilate Indigenous children into Western culture, often at the expense of eradicating their native culture, languages, and identities.