Sure! Let's create an arithmetic sequence.
Creating an Arithmetic Sequence
Let's define our arithmetic sequence as follows:
- First Term (a₁): 2
- Common Difference (d): 3
Listing the First Six Terms
Using the first term and the common difference, the sequence is:
- \( a_1 = 2 \)
- \( a_2 = a_1 + d = 2 + 3 = 5 \)
- \( a_3 = a_2 + d = 5 + 3 = 8 \)
- \( a_4 = a_3 + d = 8 + 3 = 11 \)
- \( a_5 = a_4 + d = 11 + 3 = 14 \)
- \( a_6 = a_5 + d = 14 + 3 = 17 \)
So our sequence is: 2, 5, 8, 11, 14, 17
Explanation of Arithmetic Sequence
An arithmetic sequence is defined by a constant difference between consecutive terms. In this sequence, the common difference \(d\) is 3. This means that each term is derived by adding 3 to the previous term.
Fifth Term in Subscript Notation
The fifth term of our arithmetic sequence is denoted as: \[ a_5 = 14 \]
Explicit Formula
The explicit formula for an arithmetic sequence can be written as:
\[ a_n = a_1 + (n - 1)d \]
Substituting in our values: \[ a_n = 2 + (n - 1) \cdot 3 \]
This simplifies to: \[ a_n = 3n - 1 \]
Finding the 100th Term
To find the 100th term \(a_{100}\): \[ a_{100} = 3(100) - 1 = 300 - 1 = 299 \]
Recursive Formula
The recursive formula for our arithmetic sequence will be as follows:
- Initial Value: \( a_1 = 2 \)
- Recursive Step: \( a_n = a_{n-1} + 3 \) for \( n \geq 2 \)
Graphing the Sequence
To graph the arithmetic sequence, we plot the points corresponding to each term:
- \( (1, 2) \)
- \( (2, 5) \)
- \( (3, 8) \)
- \( (4, 11) \)
- \( (5, 14) \)
- \( (6, 17) \)
- And so forth...
Here's a rough representation (not to scale):
Y
|
17| * (6, 17)
14| * (5, 14)
11| * (4, 11)
8| * (3, 8)
5|* (2, 5)
2|* (1, 2)
|_________________ X
1 2 3 4 5 6
This graph demonstrates the linear growth of the terms as we move along the x-axis. Each point signifies a term in the arithmetic sequence, and they form a straight line due to the constant difference between successive terms.