Arithmetic Sequences and Explicit Rules

Flashcards covering arithmetic sequence formulas, explicit nth term expressions, and mathematical justifications based on lecture questions 22 through 26.

Sequence Rule for Question 22

The expression $5n$ representing the $n\text{th}$ term of the sequence.

Reasoning for Rule $a_n = 5n$

Substituting $n = 1$ into $5n$ gives $5$, and each increase of $n$ by $1$ increases the term by $5$.

Explicit Rule for Increasing Sequence (Increase of $6$, First Term $8$)

The rule $a_n = 6n + 2$, which generates a first term of $8$ when $n = 1$ and increases by $6$ for each subsequent term.

Reasoning for Sequence $4, 10, 16, 22, \dots$ Rule $a_n = 6n - 2$

Substituting $n = 1$ into $6n - 2$ gives $4$, and each increase of $n$ by $1$ increases the term by $6$.

Explicit Rule for Increasing Sequence (Increase of $7$, First Term $10$)

The rule $a_n = 7n + 3$, which yields $10$ for $n = 1$ and increases by $7$ for each unit increase in $n$.