An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference.
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The general form of an arithmetic sequence can be written as $a_n = a_1 + (n-1)d$ where $a_n$ is the nth term, $a_1$ is the first term, and $d$ is the common difference.
The sum of the first $n$ terms of an arithmetic sequence (also known as an arithmetic series) can be calculated using the formula $S_n = \frac{n}{2} \left( 2a_1 + (n-1)d \right)$.
In an arithmetic sequence, if you know any two terms and their positions, you can determine the common difference and other terms in the sequence.
The common difference $d$ in an arithmetic sequence can be positive, negative, or zero.
Inserting additional terms into an existing arithmetic sequence maintains its property if each new term continues to follow the same common difference.
Review Questions
What is the formula for finding the nth term in an arithmetic sequence?
How do you calculate the sum of the first n terms in an arithmetic series?
If given two non-consecutive terms of an arithmetic sequence, how would you find the common difference?
Related terms
Geometric Sequence: A sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
Common Difference: The constant amount that each term in an arithmetic sequence differs from its preceding term.