- Define the Sequence: Decide on the mathematical formula or rule that generates the sequence.
- Generate Terms: Compute the terms of the sequence up to a reasonable number (since we can't truly generate an infinite number of terms).
- Plot the Sequence: Visualize the terms using a plot.
For demonstration, I'll consider a few common types of sequences and plot them:
- Arithmetic Sequence: A sequence in which each term after the first is obtained by adding a constant difference to the previous term. an=a+(n−1)da_n = a + (n-1)dan=a+(n−1)dwhere aaa is the first term and ddd is the common difference.
- Geometric Sequence: A sequence in which each term after the first is obtained by multiplying the previous term by a constant ratio. an=arn−1a_n = ar^{n-1}an=arn−1where aaa is the first term and rrr is the common ratio.
- Harmonic Sequence: A sequence in which each term is the reciprocal of an arithmetic sequence. an=1na_n = \frac{1}{n}an=n1
Let's plot the first 50 terms of each of these sequences.Arithmetic Sequence
- First term a=1a = 1a=1
- Common difference d=2d = 2d=2
Geometric Sequence
- First term a=1a = 1a=1
- Common ratio r=2r = 2r=2
Harmonic Sequence
- an=1na_n = \frac{1}{n}an=n1
I'll generate and plot these sequences in Python.
Here are the plots for the three different types of sequences:
- Arithmetic Sequence (an=1+(n−1)×2a_n = 1 + (n-1) \times 2an=1+(n−1)×2): The terms increase linearly.
- Geometric Sequence (an=1×2(n−1)a_n = 1 \times 2^{(n-1)}an=1×2(n−1)): The terms increase exponentially.
- Harmonic Sequence (an=1na_n = \frac{1}{n}an=n1): The terms decrease and approach zero as nnn increases.
