Examples:

• $1, 3, 5, 7, \ldots$ ✔️
• $6, 4, 2, 0, \ldots$ ✔️
• $1, 2, 4, 8, 16, \ldots$ ❌ The first term of an arithmetic sequence is denoted by '$a$', the common difference by '$d$', and the term number by '$n$'. The $n^{th}$ term is denoted by $u_n$.

Example:

• For the sequence $5, 9, 13, 17, \ldots$:
• $a = 5$ (first term)
• $d = 4$ (common difference)
• $u_3 = 13$ (third term), etc.

An arithmetic sequence can be written as:

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

Example: Consider the sequence $3, 7, 11, 15, \dots$ :

• Here, $a = 3$ and $d = 4$ .
• To find the $5th$ term ( $a_5$ ):

$a_5 = 3 + (5 - 1) \times 4 = 3 + 16 = 19$

So, the $5th$ term is $19$.

1. Sum of the First $n$ Terms (Arithmetic Series): The sum of the first $n$ terms of an arithmetic sequence (also called an arithmetic series) is given by:

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

• $S_n$ is the sum of the first $n$ terms.
• $n$ is the number of terms.
• $a$ is the first term.
• $d$ is the common difference. Alternatively, it can also be written as:

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

• Where $a_n$ is the $n$ th term.

Example: Using the previous sequence $3, 7, 11, 15, \dots$ , find the sum of the first 5 terms:

• First term $a = 3$ .
• Common difference $d = 4$ .
• Number of terms $n = 5$ .
• The $5th$ term $a_5 = 19$ . Now, using the sum formula:

$S_5 = \frac{5}{2} \left( 3 + 19 \right) = \frac{5}{2} \times 22 = 5 \times 11 = 55$

So, the sum of the first $5$ terms is $55$.

Problem: Given an arithmetic sequence with the first term $a = 2$ and the common difference $d = 5$, find the 10th term and the sum of the first 10 terms.

• Arithmetic Sequence: A sequence where each term is found by adding a constant difference to the previous term.
• nth Term: $a_n = a + (n - 1)d$ .
• Sum of First $n$ Terms: $S_n = \frac{n}{2} \left( 2a + (n - 1)d \right)$ or $S_n = \frac{n}{2} \left( a + a_n \right)$ .