Arithmetic Progressions - nth Term of an AP

Definition of an Arithmetic Progression (AP): An AP is a sequence of numbers in which each term after the first is obtained by adding a fixed number to the preceding term. Imagine a staircase where each step has the exact same height; that height represents the common difference.

First Term and Common Difference: The first term of an AP is denoted by $a$. The fixed number added to each term is called the common difference, denoted by $d$. If you plot the terms of an AP on a number line, $d$ is the constant distance between any two adjacent points.

General Form of an AP: The sequence is represented as $a, a+d, a+2d, a+3d, \dots$. Visually, this looks like a starting point $a$ followed by a series of equal 'jumps' of size $d$.

The $n^{th}$ Term Concept: The $n^{th}$ term, denoted as $a_n$, is the value located at the $n^{th}$ position in the sequence. To reach the $n^{th}$ position, you start at $a$ and perform $(n-1)$ jumps of size $d$.

Direction of the Sequence: If $d > 0$, the AP is increasing and the values move towards positive infinity. If $d < 0$, the AP is decreasing and the values move towards negative infinity. If $d = 0$, the AP is constant, appearing as a horizontal line of identical points on a graph.

Linear Relationship: If we plot the position $n$ on the x-axis and the value $a_n$ on the y-axis, all points $(n, a_n)$ will lie on a straight line. This shows that the $n^{th}$ term is a linear function of $n$.

Finite and Infinite APs: An AP that has a limited number of terms is called a finite AP and has a last term ($l$). An AP that goes on forever is called an infinite AP, represented with an ellipsis ($\dots$) at the end.