An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference, denoted by $d$. The general form of an arithmetic progression can be expressed as:

$a, a + d, a + 2d, a + 3d, \ldots$

Here, $a$ is the first term of the sequence, and $d$ is the common difference. The $n^{th}$ term of an arithmetic progression can be calculated using the formula:

$a_n = a + (n - 1)d$

Where:

• $a_n$ = $n^{th}$ term
• $a$ = First term
• $d$ = Common difference

For example, in the arithmetic progression $2, 5, 8, 11, \ldots$, the first term $a = 2$ and the common difference $d = 3$.

To find a specific term in the sequence, we can use the formula mentioned above. For instance, to find the 10th term of the sequence:

$a_{10} = 2 + (10 - 1) \cdot 3 = 2 + 27 = 29$

The sum of the first $n$ terms of an arithmetic progression, denoted as $S_n$, can be calculated using the formula:

$S_n = \frac{n}{2} \cdot (2a + (n - 1)d)$

Alternatively, the sum can also be expressed as:

$S_n = \frac{n}{2} \cdot (a + a_n)$

Where:

• $S_n$ = Sum of the first $n$ terms
• $a_n$ = $n^{th}$ term

For example, if we want to find the sum of the first 10 terms of the AP $2, 5, 8, 11, \ldots$:

We already know:

• First term $a = 2$
• Common difference $d = 3$
• 10th term $a_{10} = 29$

Using the formula for the sum:

$S_{10} = \frac{10}{2} \cdot (2 \cdot 2 + (10 - 1) \cdot 3)$

$S_{10} = 5 \cdot (4 + 27) = 5 \cdot 31 = 155

Thus, the sum of the first 10 terms of the arithmetic progression is $155$.

Arithmetic progressions have various applications in real-life scenarios, including finance (calculating interest), physics (uniform motion), and computer science (algorithms). Recognizing the structure of APs allows for efficient calculations and predictions.