A Sequence $S _ { 1 } , S _ { 2 } , S _ { 3 }$

question 5

Essay

A sequence $S _ { 1 } , S _ { 2 } , S _ { 3 }$ ... is defined recursively as follows: $$\begin{array} { l } s _ { k } = 5 s _ { k - 1 } + \left( s _ { k - 2 } \right) ^ { 2 } \quad \text { for all integers } k \geq 3 \\s _ { 1 } = 4 \\s _ { 2 } = 8\end{array}$$ Use (strong) mathematical induction to prove that sn is divisible by 4 for all integers $$n \geq 1$$

---

Definitions:

Preferred Shareholders

Investors who own preferred shares in a company, which often grant them benefits such as fixed dividend payments and priority over common shareholders in case of liquidation.

Marketable Securities

Liquid financial instruments that can be quickly converted into cash at a reasonable price.

Inventory Period

The time span between the acquisition of inventory and its sale, crucial for managing inventory efficiency and cash flow.

Credit Sales

Transactions where goods or services are sold and payment is deferred to a future date, typically involving the issuance of an invoice.