Key Points

An Arithmetic Progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a fixed number to the preceding term. This fixed number is known as the common difference.

The common difference, denoted by $d$, is the constant difference between consecutive terms in an AP. It is calculated as $d = a_{k+1} - a_k$, where $a_k$ and $a_{k+1}$ are consecutive terms. The common difference can be positive, negative, or zero.

The general form of an AP is $a, a+d, a+2d, a+3d, \ldots$, where $a$ is the first term and $d$ is the common difference. To define an AP, you only need to know the first term and the common difference.

The $n$th term (or general term) of an AP is given by the formula $a_n = a + (n-1)d$. In this formula, $a$ is the first term, $d$ is the common difference, and $n$ is the position of the term in the sequence.

To verify if a given list of numbers is an AP, calculate the difference between consecutive terms. If the difference, such as $a_2 - a_1$, $a_3 - a_2$, etc., is the same for all pairs, the sequence is an AP.

The sum of the first $n$ terms of an AP, denoted by $S_n$, is calculated using the formula $S_n = \frac{n}{2}[2a + (n-1)d]$. Here, $n$ is the number of terms, $a$ is the first term, and $d$ is the common difference.

If the first term $a$ and the last term $l$ (which is the $n$th term, $a_n$) of a finite AP are known, the sum can be found using the simpler formula $S_n = \frac{n}{2}(a+l).

The $n$th term of an AP can be found if the sum of terms is known. It is the difference between the sum of the first $n$ terms and the sum of the first $(n-1)$ terms: $a_n = S_n - S_{n-1}$.

The sum of the first $n$ positive integers ($1, 2, 3, \ldots, n$) is a special case of an AP where $a=1$ and $d=1$. The sum is given by the direct formula $S_n = \frac{n(n+1)}{2}$.

If three numbers $a, b, c$ are in an AP, then the middle term $b$ is called the arithmetic mean of $a$ and $c$. It is calculated as $b = \frac{a+c}{2}$.