Types of Sequences

Constant First Difference: A linear sequence, also known as an arithmetic progression, is characterized by a common difference ($d$) that is added or subtracted consistently between consecutive terms.

Growth Pattern: Because the change is constant, the terms grow or decay at a steady, linear rate. If you were to plot the term values against their positions, they would form a straight line.

General Form: The $n^{th}$ term of a linear sequence is always in the form $an + b$, where $a$ represents the common difference and $b$ is a constant adjusted to fit the first term.

Constant Second Difference: Unlike linear sequences, the first differences in a quadratic sequence change. However, the differences between those differences (the second differences) remain constant.

Mathematical Foundation: These sequences are related to square numbers ($n^2$). If the second difference is $k$, the $n^{th}$ term formula will involve a leading term of $\frac{k}{2}n^2$.

Identification: To identify a quadratic sequence, calculate the differences between terms twice. If the first row of differences is $3, 5, 7, 9$ and the second row is $2, 2, 2, 2$, the sequence is quadratic.

Common Ratio ($r$): In a geometric sequence (or progression), each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This ratio can be an integer, a fraction, or even a negative number.

Exponential Growth/Decay: Geometric sequences exhibit exponential behavior. If $|r| > 1$, the terms grow rapidly; if $0 < |r| < 1$, the terms shrink toward zero.

Formulaic Structure: The term-to-term rule is expressed as $a_{n+1} = r \cdot a_n$. To find the common ratio from a list of terms, divide any term by its immediate predecessor ($r = \frac{a_{n+1}}{a_n}$).

Fibonacci Sequences: A recursive sequence where each term is the sum of the two preceding terms. The classic Fibonacci sequence starts $1, 1, 2, 3, 5, 8...$, but any sequence following the rule $a_n = a_{n-1} + a_{n-2}$ is considered Fibonacci-type.

Triangular Numbers: These represent the number of dots needed to form an equilateral triangle. The sequence is $1, 3, 6, 10, 15...$, where the $n^{th}$ term is the sum of integers from $1$ to $n$.

Cube and Square Numbers: These are simple sequences where the $n^{th}$ term is simply $n^3$ or $n^2$. They often serve as the building blocks for more complex quadratic or cubic sequences.

Linear vs. Geometric: Linear sequences change by adding a fixed amount, while geometric sequences change by multiplying by a fixed amount. Linear growth is steady, whereas geometric growth accelerates.

Linear vs. Quadratic: Linear sequences have a constant rate of change. Quadratic sequences have a rate of change that itself changes at a constant rate.

The Difference Test: Always start by finding the first differences. If they are constant, it is linear. If they are not, find the second differences; if those are constant, it is quadratic.

The Ratio Test: If neither the first nor second differences are constant, check if there is a common multiplier by dividing consecutive terms. If the ratio is the same, it is geometric.

Check for Fibonacci: If the sequence seems to grow quickly but isn't geometric, check if adding two terms gives the third. This is a common 'hidden' pattern in exam questions.

Verification: Once you find an $n^{th}$ term formula, always test it with $n=1$, $n=2$, and $n=3$ to ensure it produces the correct terms of the sequence.