Arithmetic Sequences
Definition: An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This difference is known as the common difference.
Common Difference: Let \(d\) be the common difference. If \(a_1\) is the first term of the sequence, then each term can be expressed as:
- \(a_2 = a_1 + d\)
- \(a_3 = a_2 + d = a_1 + 2d\)
- and so on.
Explicit Formula: The explicit formula for an arithmetic sequence is given by:
\[ a_n = a_1 + (n-1)d \]
where:
- \(a_n\) is the \(n\)-th term,
- \(a_1\) is the first term,
- \(d\) is the common difference,
- \(n\) is the term number.
Recursive Formula: The recursive formula for an arithmetic sequence can be expressed as:
\[ \begin{align*} a_1 & = \text{(first term)} \ a_n & = a_{n-1} + d \quad \text{for } n > 1 \end{align*} \]
Graph of an Arithmetic Sequence: The graph of an arithmetic sequence is a straight line when plotted on a Cartesian coordinate system. This is because there is a constant change (the common difference) in the values of the sequence with respect to the term number.
Example:
Let’s consider an arithmetic sequence where the first term \(a_1 = 3\) and the common difference \(d = 2\).
- The sequence would be: \[ 3, 5, 7, 9, 11, \ldots \]
Explicit Formula: \[ a_n = 3 + (n-1) \cdot 2 = 2n + 1 \]
Recursive Formula: \[ \begin{align*} a_1 & = 3 \ a_n & = a_{n-1} + 2 \quad \text{for } n > 1 \end{align*} \]
Graph: To illustrate, we can plot the sequence on a graph:
- \(x\) (horizontal axis) represents \(n\) (term number),
- \(y\) (vertical axis) represents \(a_n\) (value of the term).
|
11 | * (6)
10 | *
9 | *
8 | *
7 | *
6 |
5 | *
4 |
3 | * (1)
2 |
1 |
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1 2 3 4 5 6
- Each point corresponds to a term in the sequence, demonstrating the linearity and equal spacing between the points, reflective of the constant difference of \(2\).