Mathematics Sequences and Patterns Lecture Notes

A collection of vocabulary and concepts regarding mathematical sequences, including arithmetic and geometric progressions, pattern recognition, and practical application problems.

Arithmetic Sequence

A sequence of numbers where the difference between consecutive terms is constant, such as in the example $48, 41, 34, 27, 20, \dots$ where the common difference is $-7$.

Geometric Sequence

A sequence of numbers where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio, as seen in $2, 10, 50, 250, 1250, \dots$ with a ratio of $5$.

Jack's Running Pattern

A linear sequence starting at $1\,mile$ in Week 1 and increasing by $0.5\,miles$ each subsequent week ($1.5\,miles$ in Week 2, $2\,miles$ in Week 3).

Rule for the nth term

A formula used to find the value of any term in a sequence based on its position $n$; for the sequence $8, 3, -2, -7, -12, -17$, the rule is $a_n = -5n + 13$.

Gabriel Sequence

A repeating pattern of the 7-letter name "Gabriel" used to determine the value of a term at a specific position, such as the $1000th$ letter.

One Step Forward, Three Steps Back Pattern

A movement pattern starting at $0$ on a number line where a person moves $+1$ and then $-3$; after $8$ steps (4 full cycles of $-2$ steps each), the position is $-8$.

Arithmetic Mean of $-8$ and $-18$

The second term in a three-term arithmetic sequence starting with $-8$ and ending with $-18$, calculated as $-13$.

Geometric Mean of $-8$ and $-18$

The second term in a three-term geometric sequence starting with $-8$ and ending with $-18$, which results in a term of either $12$ or $-12$ depending on the common ratio ($\pm 1.5$).

Common Ratio

The constant value by which a term is multiplied to get the next term in a geometric sequence; for example, the ratio is $3$ if the first term is $1$ and the subsequent terms are $3, 9, 27, 81$.

Area of a Rectangle

The mathematical product of the width $w$ and length $l$, represented by expressions such as $(l \times w)$ or components that sum to the total area.

Paper Folding Thickness

An exponential growth pattern where a sheet of paper $0.004\,inches$ thick doubles in thickness with each fold ($0.008\,inches$ after 1 fold, $0.016\,inches$ after 2 folds).

Meet-and-Greet Conversations

The number of unique two-person interactions in a group of size $n$, calculated using the expression $\frac{n(n-1)}{2}$; for $3$ people, there are $3$ conversations.

Email Addresses Exchanged

The total number of email addresses shared when everyone in a group of $n$ people exchanges info with everyone else, calculated as $n(n-1)$; for $3$ people, $6$ addresses are exchanged.