An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference.
Key Properties of an Arithmetic Sequence
Common Difference
The common difference, often denoted as $d$, is the amount added to each term to get the next term in the sequence. For example, in the sequence 2, 5, 8, 11, …, the common difference $d$ is 3.
General Formula
The $n$-th term of an arithmetic sequence can be found using the formula:
$a_n = a_1 + (n – 1) times d$
where $a_n$ is the $n$-th term, $a_1$ is the first term, and $d$ is the common difference.
Example
Consider the arithmetic sequence 4, 7, 10, 13, …
- The first term $a_1$ is 4.
- The common difference $d$ is 3.
Using the formula, the 5th term would be:
$a_5 = 4 + (5 – 1) times 3 = 4 + 12 = 16$
Sum of an Arithmetic Sequence
The sum of the first $n$ terms of an arithmetic sequence, known as the arithmetic series, can be calculated using the formula:
$S_n = frac{n}{2} times (2a_1 + (n – 1) times d)$
or equivalently,
$S_n = frac{n}{2} times (a_1 + a_n)$
Example
Let’s find the sum of the first 5 terms of the sequence 4, 7, 10, 13, 16.
- The first term $a_1$ is 4.
- The fifth term $a_5$ is 16.
Using the sum formula:
$S_5 = frac{5}{2} times (4 + 16) = frac{5}{2} times 20 = 50$
Applications
Arithmetic sequences are used in various real-world scenarios, such as calculating monthly savings, planning schedules, and even in certain computer algorithms. Understanding these sequences helps in solving problems involving regular intervals.
Conclusion
Arithmetic sequences are a fundamental concept in mathematics, characterized by their constant common difference. They have straightforward formulas for finding terms and sums, making them easy to work with and apply in various contexts.