An arithmetic sequence (or arithmetic progression) is a sequence of numbers in which each term after the first is obtained by adding a constant difference, called the common difference (denoted as $d$), to the previous term.

Key Concepts in Arithmetic Sequences

1. General Form of an Arithmetic Sequence: $a, a + d, a + 2d, a + 3d, \dots$ Here:

   • $a$ is the first term of the sequence.
   • $d$ is the common difference between consecutive terms.

2. Finding the $n$-th Term ($a_n$) of an Arithmetic Sequence: The $n$-th term (or general term) of an arithmetic sequence can be found using the formula: $a_n = a + (n - 1) \times d$

   • $a_n$: The $n$-th term of the sequence
   • $a$: The first term
   • $d$: The common difference
   • $n$: The position of the term in the sequence

3. Sum of the First $n$ Terms ( $S_n$ ) of an Arithmetic Sequence: To find the sum of the first $n$ terms, use the formula: $S_n = \frac{n}{2} \times (2a + (n - 1) \times d)$ or equivalently: $S_n = \frac{n}{2} \times (a + a_n)$ where:

   • $S_n$: Sum of the first $n$ terms
   • $a$: First term
   • $d$: Common difference
   • $a_n$: $n$-th term

4. Common Difference: The common difference $d$ can be found if you know two consecutive terms. If $a_2$ and $a_1$ are two consecutive terms: $d = a_2 - a_1$

Example

Suppose you have the arithmetic sequence: $3, 7, 11, 15, \dots$.

• The first term $a = 3$.
• The common difference $d = 7 - 3 = 4$.

To find the 5th term ($a_5$): $a_5 = a + (5 - 1) \times d = 3 + 4 \times 4 = 3 + 16 = 19$ So, the 5th term is 19.

If you want to find the sum of the first 5 terms ($S_5$):

= \frac{5}{2} \times (6 + 16) = \frac{5}{2} \times 22 = 55$$ So, the sum of the first 5 terms is 55. --- Would you like more examples or explanations? ### Follow-Up Questions: 1. How can you determine if a sequence is arithmetic? 2. How do you calculate the common difference if given only the first and last terms? 3. How does the formula for the sum of terms change if the sequence is infinite? 4. Can an arithmetic sequence have a negative common difference? If so, what does that mean? 5. How does an arithmetic sequence differ from a geometric sequence? ### Tip: When solving problems with sequences, identifying the first term and common difference is the key to setting up any formula correctly.