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 $$s _ { n } = \frac { 5 ^ { n + 1 } - 1 } { 4 } \quad \text { for all integers } n \geq 0 \text {. }$$

---

Definitions:

Common Fixed Expenses

Overhead costs shared by multiple business units or product lines, which are not attributable to any single unit or product.

Break-even

The point at which total revenues equal total expenses, resulting in no net loss or gain.

Sales Dollars

The total revenue generated from the sale of goods or services, measured in monetary terms.

Seed Division

In a corporate setting, the Seed Division could refer to a segment of a company focused on developing and selling seeds or a startup phase within an organization aimed at growing new business ideas.