create your own rule or formula in arithmetic sequence, involving variables

Sure, let's create a custom rule for an arithmetic sequence involving variables. We'll define a general term formula and then derive some specific properties from it.

### General Term Formula of an Arithmetic Sequence

In a standard arithmetic sequence, each term is obtained by adding a constant difference \(d\) to the previous term. We'll define the \(n\)-th term of our arithmetic sequence as:

\[ a_n = a_1 + (n-1)d \]

where:
- \(a_n\) is the \(n\)-th term,
- \(a_1\) is the first term,
- \(d\) is the common difference,
- \(n\) is the term number.

### Custom Rule: Incorporating Variables

Let's introduce variables to create a custom rule. We'll use a linear combination of the first term and the common difference, controlled by an additional variable \(k\). 

#### Rule: Custom Arithmetic Sequence

Define the \(n\)-th term of the arithmetic sequence with a variable \(k\) as:

\[ a_n = a_1 + k(n-1)d \]

where:
- \(a_n\) is the \(n\)-th term,
- \(a_1\) is the first term,
- \(d\) is the common difference,
- \(n\) is the term number,
- \(k\) is a scaling variable.

### Properties and Examples

1. **Standard Arithmetic Sequence**: When \(k = 1\),
   \[ a_n = a_1 + (n-1)d \]

2. **Modified Arithmetic Sequence**: When \(k = 2\),
   \[ a_n = a_1 + 2(n-1)d \]
   This sequence grows twice as fast as the standard sequence.

3. **Variable Control**: For \(k = \frac{1}{2}\),
   \[ a_n = a_1 + \frac{1}{2}(n-1)d \]
   This sequence grows half as fast as the standard sequence.

### Derivation

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

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

\[ S_n = \sum_{i=1}^{n} a_i \]
\[ S_n = \sum_{i=1}^{n} \left( a_1 + k(i-1)d \right) \]
\[ S_n = na_1 + kd \sum_{i=1}^{n} (i-1) \]
\[ S_n = na_1 + kd \left( \frac{(n-1)n}{2} \right) \]
\[ S_n = na_1 + \frac{kd(n-1)n}{2} \]

### Example

For a sequence with \(a_1 = 3\), \(d = 2\), and \(k = 1.5\):

1. First term (\(n=1\)):
   \[ a_1 = 3 \]
2. Second term (\(n=2\)):
   \[ a_2 = 3 + 1.5 \cdot (2-1) \cdot 2 = 3 + 3 = 6 \]
3. Third term (\(n=3\)):
   \[ a_3 = 3 + 1.5 \cdot (3-1) \cdot 2 = 3 + 6 = 9 \]

So the sequence is: \( 3, 6, 9, \ldots \)

Would you like more details or have any questions?

### Additional Questions:

1. How does changing the value of \(k\) affect the growth rate of the sequence?
2. What happens if \(k\) is a negative value?
3. How do you find the common difference \(d\) if only the first term and some other term are given?
4. Can this custom rule be applied to geometric sequences?
5. How would the sum formula change if \(k\) were a variable function of \(n\)?
6. What are the implications of setting \(k = 0\)?
7. How would you graph this custom arithmetic sequence?
8. How does this sequence compare to a quadratic sequence?

### Tip:

When dealing with sequences, always verify the first few terms to ensure your formula is applied correctly.

Learn how to create custom rules in arithmetic sequences involving variables. Explore the general term formula and its variations using scaling variables. Suitable for advanced high school students and those interested in mathematical sequences.

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