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

question 22

Essay

A sequence $a _ { 0 } , a _ { 1 } , a _ { 2 } , \ldots$ satisfies the recurrence relation $a _ { k } = 4 a _ { k - 1 } - 3 a _ { k - 2 }$ with initial conditions $a _ { 0 } = 1$ and $a _ { 1 } = 2$ . Find an explicit formula for the sequence.

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Definitions:

Reserve Requirements

The minimum amount of funds that a bank is required to hold in reserve, determined by central banking authorities, to ensure that the institution remains liquid.

Monetary Control

Monetary Control involves the regulation of the money supply and interest rates by central banks to manage economic stability and growth.

Open Market Operations

The buying and selling of government securities by a central bank in order to control the money supply and influence interest rates.

Excess Reserves

The capital reserves held by a bank or financial institution in excess of what is required by regulators, central bank, or other governing body.