Sure, let's create a rule involving an algebraic sequence where each term is a monomial.

### General Term Formula for a Monomial Sequence

In a monomial sequence, each term can be represented as a power of the term number multiplied by a constant coefficient. We'll define the \(n\)-th term of our monomial sequence as:

\[ a_n = c \cdot n^k \]

where:
- \(a_n\) is the \(n\)-th term,
- \(c\) is a constant coefficient,
- \(n\) is the term number,
- \(k\) is the exponent.

### Custom Rule: Monomial Sequence

Let's define a custom rule for a monomial sequence using variables \(c\) and \(k\):

#### Rule: Custom Monomial Sequence

\[ a_n = c \cdot n^k \]

### Properties and Examples

1. **Linear Sequence**: When \(k = 1\),
   \[ a_n = c \cdot n \]
   This is a linear sequence with each term increasing by the coefficient \(c\).

2. **Quadratic Sequence**: When \(k = 2\),
   \[ a_n = c \cdot n^2 \]
   This is a quadratic sequence with each term being a square of the term number, scaled by \(c\).

3. **Cubic Sequence**: When \(k = 3\),
   \[ a_n = c \cdot n^3 \]
   This is a cubic sequence with each term being a cube of the term number, scaled by \(c\).

### Example

For a sequence with \(c = 2\) and \(k = 3\):

1. First term (\(n=1\)):
   \[ a_1 = 2 \cdot 1^3 = 2 \]
2. Second term (\(n=2\)):
   \[ a_2 = 2 \cdot 2^3 = 2 \cdot 8 = 16 \]
3. Third term (\(n=3\)):
   \[ a_3 = 2 \cdot 3^3 = 2 \cdot 27 = 54 \]

So the sequence is: \( 2, 16, 54, \ldots \)

### Sum of the First \(n\) Terms

The sum \(S_n\) of the first \(n\) terms of this custom monomial sequence can be derived as follows:

For \( k = 1 \):
\[ S_n = \sum_{i=1}^{n} (c \cdot i) = c \cdot \sum_{i=1}^{n} i = c \cdot \frac{n(n+1)}{2} \]

For \( k = 2 \):
\[ S_n = \sum_{i=1}^{n} (c \cdot i^2) = c \cdot \sum_{i=1}^{n} i^2 = c \cdot \frac{n(n+1)(2n+1)}{6} \]

For \( k = 3 \):
\[ S_n = \sum_{i=1}^{n} (c \cdot i^3) = c \cdot \sum_{i=1}^{n} i^3 = c \cdot \left(\frac{n(n+1)}{2}\right)^2 \]

Would you like more details or have any questions?

### Additional Questions:

1. How does the sequence change as the exponent \(k\) increases?
2. What happens if the coefficient \(c\) is negative?
3. How do you determine the sum formula for an arbitrary exponent \(k\)?
4. Can this rule be applied to fractional exponents?
5. How does this sequence compare to an exponential sequence?
6. What are the implications of setting \(c = 0\)?
7. How would you graph this custom monomial sequence for different values of \(k\)?
8. How does the sequence behave as \(n\) approaches infinity for different values of \(k\)?

### Tip:

When working with sequences involving powers, always check a few initial terms to ensure the sequence behaves as expected for the given coefficient and exponent.

Explore how to create rules for monomial sequences in algebra. Understand the general term formula, custom rules for different exponents, and examples of linear, quadratic, and cubic sequences. Suitable for advanced high school students.

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