A Sequence Is Defined Recursively as Follows $$s _ { k } = 5 s _ { k - 1 } + 1$$

question 7

Essay

A sequence is defined recursively as follows: $$s _ { k } = 5 s _ { k - 1 } + 1$$ for all integers $k \geq 1$
$$s _ { 0 } = 1 \text {. }$$
Use mathematical induction to verify that this sequence satisfies the explicit formula $$s _ { n } = \frac { 5 ^ { n + 1 } - 1 } { 4 } \quad \text { for all integers } n \geq 0 \text {. }$$

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Definitions:

Labor-Leisure Choices

The decision-making process that individuals use to allocate their time between working to earn money and enjoying leisure activities.

Equilibrium Wage Rate

The compensation level where labor availability matches labor needs.

Profit-Maximizing

A strategy or approach aimed at achieving the highest possible profit.

Perfectly Competitive

A market structure characterized by many small firms, homogeneous products, and easy market entry and exit, leading to minimal profit margins for the firms.