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A Sequence Is Defined Recursively as Follows a0=2 and ak=4ak1+1 for all k1.a _ { 0 } = 2 \quad \text { and } \quad a _ { k } = 4 a _ { k - 1 } + 1 \quad \text { for all } k \geq 1 .

question 25

Essay

A sequence is defined recursively as follows: a0=2 and ak=4ak1+1 for all k1.a _ { 0 } = 2 \quad \text { and } \quad a _ { k } = 4 a _ { k - 1 } + 1 \quad \text { for all } k \geq 1 .
It is proposed that an explicit formula for this sequence is an=74n13a _ { n } = \frac { 7 \cdot 4 ^ { n } - 1 } { 3 }
Use mathematical induction to check whether this proposed formula is correct.

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Serious Pollutant

Substances that cause pollution and can have significant harmful effects on the environment, human health, or both.

Accumulates

The process of gathering or increasing in quantity over time, often used in the context of data, wealth, or substances.

Excreted

The process of eliminating waste materials from the body or a biological system, crucial for maintaining homeostasis and health.

Material Flow Economy

A concept that emphasizes the importance of managing the flow of materials through the economy to minimize waste and resource depletion.

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