An arithmetic sequence is a series of numbers in which the difference between consecutive terms is constant. This constant difference is known as the common difference.
Understanding Arithmetic Sequences
An arithmetic sequence is a list of numbers where each term after the first is found by adding a constant, called the common difference, to the previous term. For example, in the sequence 3, 7, 11, 15, …, the common difference is 4.
Formula for the Common Difference
The common difference ($d$) can be found using the formula:
$d = a_{n+1} – a_n$
where $a_{n+1}$ is the $(n+1)$-th term and $a_n$ is the $n$-th term.
Example Calculation
Consider the sequence: 5, 9, 13, 17, …
- Identify two consecutive terms: $a_2 = 9$ and $a_1 = 5$
- Apply the formula: $d = a_2 – a_1 = 9 – 5 = 4$
Thus, the common difference is 4.
Verifying the Common Difference
To ensure the common difference is correct, you can check multiple pairs of consecutive terms:
- $a_3 – a_2 = 13 – 9 = 4$
- $a_4 – a_3 = 17 – 13 = 4$
Since the difference is consistent, the common difference is verified.
Practical Applications
Understanding the common difference helps in solving various problems, such as finding the $n$-th term of a sequence. The $n$-th term ($a_n$) can be calculated using the formula:
$a_n = a_1 + (n-1)d$
For example, to find the 10th term of the sequence 5, 9, 13, 17, …:
- Identify $a_1 = 5$ and $d = 4$
- Apply the formula: $a_{10} = 5 + (10-1) times 4 = 5 + 36 = 41$
So, the 10th term is 41.
Conclusion
Finding the common difference in an arithmetic sequence is straightforward and essential for understanding the sequence’s structure. By subtracting any term from the subsequent term, you can determine the common difference and use it to solve further problems involving the sequence.