Key Points

A sequence is an ordered list of numbers, where each number is called a term. We use notation like $t_n$ to represent the term in the $n^{\text{th}}$ position.

An explicit rule or formula defines the $n^{\text{th}}$ term of a sequence using its position number, $n$. For example, the sequence of even numbers can be written as $t_n = 2n$.

A recursive rule defines a term of a sequence using one or more preceding terms. It always requires one or more initial terms to be stated. For example, $t_1 = 5, t_n = t_{n-1} + 2$.

An Arithmetic Progression (AP) is a sequence where each term is obtained by adding a fixed number, called the common difference ($d$), to the previous term. For example, $3, 7, 11, 15, \ldots$ is an AP with $d=4$.

The explicit formula for the $n^{\text{th}}$ term of an AP is $t_n = a + (n-1)d$. Here, $a$ is the first term, $d$ is the common difference, and $n$ is the term number.

The common difference $d$ in an AP is the difference between any two consecutive terms. It is calculated as $d = t_n - t_{n-1}$. The difference can be positive, negative, or zero.

A Geometric Progression (GP) is a sequence where each term is obtained by multiplying the previous term by a fixed non-zero number, called the common ratio ($r$). For example, $2, 6, 18, 54, \ldots$ is a GP with $r=3$.

The explicit formula for the $n^{\text{th}}$ term of a GP is $t_n = ar^{n-1}$. Here, $a$ is the first term, $r$ is the common ratio, and $n$ is the term number.

The common ratio $r$ in a GP is the ratio of any two consecutive terms. It is calculated as $r = \frac{t_n}{t_{n-1}}$. The ratio can be an integer, a fraction, or a negative number.

The sum of the first $n$ natural numbers, denoted by $S_n$, is given by the formula $S_n = \frac{n(n+1)}{2}$. This formula is useful for summing long lists of consecutive numbers.

The sequence of triangular numbers is $1, 3, 6, 10, 15, \ldots$. The $n^{\text{th}}$ triangular number is the sum of the first $n$ natural numbers, so its formula is $t_n = \frac{n(n+1)}{2}$.

Each term in the square number sequence ($1, 4, 9, 16, \ldots$) is the sum of consecutive odd numbers. For example, the 4th square number is $16 = 1+3+5+7$.

This is a famous recursive sequence defined by $V_1=1, V_2=2$, and $V_n = V_{n-1} + V_{n-2}$ for $n \geq 3$. Each term is the sum of the two preceding terms, resulting in the sequence $1, 2, 3, 5, 8, \ldots$.

When the terms of an AP are plotted on a graph with the term number $n$ on the x-axis and the term value $t_n$ on the y-axis, the points $(n, t_n)$ lie on a straight line. This shows a linear growth or decay.

When the terms of a GP are plotted on a graph, the points $(n, t_n)$ lie on a curve, not a straight line. This represents exponential growth (if $|r| > 1$) or decay (if $|r| < 1$).