Sequences and Series - Arithmetic Progression (A.P.)

Definition of Arithmetic Progression (A.P.): An A.P. 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, if you plot the term number $n$ on the horizontal axis and the value of the term $a_n$ on the vertical axis, the points lie on a straight line, similar to a linear graph $y = mx + c$ where the slope $m$ corresponds to the common difference $d$.

Common Difference ($d$): The common difference is calculated as $d = a_{n} - a_{n-1}$. It can be positive, negative, or zero. If $d > 0$, the sequence is increasing (like a rising staircase); if $d < 0$, the sequence is decreasing (like a descending staircase); if $d = 0$, the sequence is constant.

General Term ($n$-th term): The $n$-th term of an A.P. with first term $a$ and common difference $d$ is denoted by $a_n$. It represents the value at the $n$-th position. In a list of terms, this is the rule that allows you to find any term without listing all previous ones.

Sum of First $n$ Terms ($S_n$): This represents the total value obtained by adding the first $n$ terms of the progression. Conceptually, this sum is equivalent to the average of the first and last terms multiplied by the number of terms. If you visualize the terms as bars of increasing height, the sum is the total area of all bars.

Arithmetic Mean (A.M.): Given two numbers $a$ and $b$, their Arithmetic Mean is a number $A$ such that $a, A, b$ are in A.P. Geometrically, $A$ is the midpoint between $a$ and $b$ on a number line. If $n$ numbers $A_1, A_2, ..., A_n$ are inserted between $a$ and $b$ such that the resulting sequence is an A.P., these are called $n$ arithmetic means.

Properties of A.P.: If a constant is added to, subtracted from, multiplied by, or divided into (except zero) each term of an A.P., the resulting sequence is also an A.P. For example, shifting a 'staircase' up or down by a fixed height does not change the equal spacing between the steps.

Selection of Terms: For solving problems involving the sum or product of terms, it is often convenient to select terms symmetrically. For three terms in A.P., use $(a-d), a, (a+d)$. For four terms, use $(a-3d), (a-d), (a+d), (a+3d)$.