๐ Understanding Explicit Formulas for Sequences
Explicit formulas are a powerful tool for defining sequences because they allow you to directly calculate any term in the sequence without needing to know the previous terms. They provide a 'shortcut' to finding, say, the 100th term without having to calculate the first 99 terms.
๐งฎ Definition of Arithmetic Sequences
An arithmetic sequence is a sequence where the difference between consecutive terms is constant. This constant difference is called the common difference, often denoted as $d$.
- ๐ Example: The sequence 2, 5, 8, 11, ... is an arithmetic sequence with a common difference of 3.
- โ๏ธ Explicit Formula: The explicit formula for an arithmetic sequence is: $a_n = a_1 + (n - 1)d$, where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.
๐ Definition of Geometric Sequences
A geometric sequence is a sequence where the ratio between consecutive terms is constant. This constant ratio is called the common ratio, often denoted as $r$.
- ๐ Example: The sequence 3, 6, 12, 24, ... is a geometric sequence with a common ratio of 2.
- โ๏ธ Explicit Formula: The explicit formula for a geometric sequence is: $a_n = a_1 * r^{(n - 1)}$, where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the term number, and $r$ is the common ratio.
๐ Arithmetic vs. Geometric: A Side-by-Side Comparison
| Feature |
Arithmetic Sequence |
Geometric Sequence |
| Definition |
Constant difference between terms. |
Constant ratio between terms. |
| Common Value |
Common Difference ($d$) |
Common Ratio ($r$) |
| Explicit Formula |
$a_n = a_1 + (n - 1)d$ |
$a_n = a_1 * r^{(n - 1)}$ |
| Operation |
Addition/Subtraction |
Multiplication/Division |
| Example Sequence |
1, 4, 7, 10, ... ($d = 3$) |
2, 6, 18, 54, ... ($r = 3$) |
๐ Key Takeaways
- โ Arithmetic sequences involve repeated addition or subtraction, while geometric sequences involve repeated multiplication or division.
- โ The explicit formulas for arithmetic and geometric sequences reflect these different operations. Notice the addition in the arithmetic formula versus the multiplication and exponentiation in the geometric formula.
- โ๏ธ Understanding the difference between the common difference ($d$) and the common ratio ($r$) is crucial for correctly applying the explicit formulas.