Types of Mathematical Sequences

Linear sequences change by a constant amount each time, known as the common difference ($d$). Because the change is constant, the first differences between terms are always equal.

The general formula for a linear sequence is $n^{th} \text{ term} = dn + b$, where $d$ is the common difference and $b$ is the 'zero term' (the value the sequence would have at $n=0$).

To find $b$, you can subtract the common difference from the first term ($a_1$). For example, if a sequence starts at $10$ and increases by $3$ each time, $d=3$ and $b = 10 - 3 = 7$, resulting in the formula $3n + 7$.

Quadratic sequences do not have a constant first difference; instead, their second differences (the difference between the differences) are constant. This indicates the formula involves an $n^2$ term.

Square numbers ($1, 4, 9, 16...$) follow the rule $n^2$, while cube numbers ($1, 8, 27, 64...$) follow $n^3$.

Triangular numbers ($1, 3, 6, 10...$) are generated by adding consecutive integers. Visually, these numbers can be arranged into equilateral triangles, where each new row adds one more dot than the previous row.

A geometric sequence is formed by multiplying the previous term by a fixed, non-zero number called the common ratio ($r$). This results in exponential growth or decay rather than linear change.

To find the common ratio, divide any term by its immediate predecessor ($r = a_{n} \div a_{n-1}$). If the ratio is greater than $1$, the sequence grows; if it is between $0$ and $1$, the sequence shrinks.

A Fibonacci sequence is created by adding the two preceding terms to find the next one ($a_n = a_{n-1} + a_{n-2}$). While the famous version starts $1, 1, 2, 3, 5...$, any sequence using this additive logic is considered Fibonacci-style.

Linear vs. Geometric: Linear sequences grow at a steady rate (straight line), whereas geometric sequences grow at an accelerating rate (curved line).

Term-to-term vs. Position-to-term: Use term-to-term for finding the very next number; use position-to-term ($n^{th}$ term) to find distant values like the $100^{th}$ term without calculating all intermediate steps.

Verification: To check if a specific number belongs to a sequence, set the $n^{th}$ term formula equal to that number and solve for $n$. If $n$ is a positive integer, the number is in the sequence; if $n$ is a fraction or decimal, it is not.

Finding the Formula: Always check the first differences first. If they are constant, it is linear. If not, check the second differences; if those are constant, it is quadratic.

Common Pitfall: When calculating the $n^{th}$ term for a decreasing linear sequence, the common difference $d$ will be negative. Ensure you include the negative sign in your formula (e.g., $-5n + 20$).