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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.


Definitions:

Psychological Intensity

The degree of strength or force of psychological phenomena, such as emotions, motivations, or experiences.

Stimulus Intensity

The magnitude or strength of a stimulus, which can influence the reaction or response it elicits.

Logarithm

The power to which a base number must be raised to produce a given number.

Fechner's Law

A principle in psychophysics stating that the perceived intensity of a stimulus changes in proportion to the logarithm of the actual stimulus intensity.

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