A Sequence $d _ { 1 } , d _ { 2 } , d _ { 3 } , \ldots$

question 9

Essay

A sequence $d _ { 1 } , d _ { 2 } , d _ { 3 } , \ldots$ satisfies the recurrence relation $d _ { k } = 8 d _ { k - 1 } - 16 d _ { k - 2 }$ with initial conditions $d _ { 1 } = 0$ and $d _ { 2 } = 1$ . Find an explicit formula for the sequence.

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Definitions:

Merchants

Individuals or businesses engaged in the selling of goods, services, or commodities for profit.

Reasonably Fit

A standard in commercial transactions that signifies goods provided under a contract must be of average or suitable quality for the purposes for which such goods are normally used.

Implied Warranty

A legal term for guarantees that are not expressly stated but are assumed in the sale of goods or services, ensuring basic standards of quality and functionality.

Merchantability

A legal concept referring to the implicit guarantee that a product is fit for the purposes for which it is sold.