Algebra - Arithmetic Progression (AP)

Definition of Arithmetic Progression (AP): An Arithmetic Progression is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant is called the common difference ($d$). Visually, you can imagine an AP as a ladder where each rung is placed at an equal vertical distance from the previous one.

First Term and Common Difference: The first term of an AP is denoted by $a$. The common difference $d$ can be positive (resulting in an increasing sequence like a rising staircase), negative (a decreasing sequence like a descending staircase), or zero (a constant sequence where all terms are identical).

General Term ($n^{th}$ term): The $n^{th}$ term, denoted as $a_n$ or $T_n$, allows you to calculate the value of any position in the sequence. Conceptually, this is like starting at the first point on a number line and taking $n-1$ equal-sized jumps of length $d$.

Sum of $n$ Terms ($S_n$): This represents the total of the first $n$ terms of the AP. Visually, if you represent each term as a vertical column of blocks, the sum is the total number of blocks used, which forms a shape similar to a trapezoid.

Arithmetic Mean: When three numbers $a, b, c$ are in AP, the middle term $b$ is called the arithmetic mean of $a$ and $c$. Geometrically, on a 1D number line, $b$ is the exact midpoint between the points $a$ and $c$, calculated as $b = \frac{a+c}{2}$.

Properties of AP: If a constant is added to, subtracted from, multiplied by, or divided into (except by zero) each term of an AP, the resulting sequence remains an AP. This reflects a uniform transformation across the entire linear pattern of the sequence.

Selection of Terms: For solving problems involving a specific number of terms in AP, it is strategically useful to pick them symmetrically. For 3 terms, use $(a-d), a, (a+d)$; for 4 terms, use $(a-3d), (a-d), (a+d), (a+3d)$. This visual symmetry ensures that the sum of terms eliminates the variable $d$.