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 $$n \geq 1$$

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Definitions:

Supreme Court

The highest judicial body in a country, typically having the ultimate authority over all other courts and adjudicating in the final appeal cases.

Reinforcer

Any stimulus that strengthens or increases the frequency or likelihood of a behavior when it occurs.

Biological Need

Fundamental requirements that organisms need to maintain life, such as food, water, and shelter.

Probability

A measure of how likely it is that some event will occur; a number expressing the ratio of favorable outcomes to the total number of possible outcomes.