Solve this sequence:

a+b = 0: 0.9, 2.25, 4.5
a+b = 1: 2.25, 4.5, 9
a+b = 2: 4.5, 9, 18
a+b = 3: 9, 18, 36
a+b = 4: 18, 36, 72
a+b = 5: 27, 54, 108
a+b = 6: 36, 72, 144

To solve this sequence, let's analyze the patterns and deduce the formula governing the sequence.

### Observations:

The sequences for each \( a + b \) are as follows:

- \( a + b = 0 \): \( 0.9, 2.25, 4.5 \)
- \( a + b = 1 \): \( 2.25, 4.5, 9 \)
- \( a + b = 2 \): \( 4.5, 9, 18 \)
- \( a + b = 3 \): \( 9, 18, 36 \)
- \( a + b = 4 \): \( 18, 36, 72 \)
- \( a + b = 5 \): \( 27, 54, 108 \)
- \( a + b = 6 \): \( 36, 72, 144 \)

### Step 1: Identify the Multiplication Factor

Looking at the ratios of consecutive terms within each sequence:

- For \( a + b = 0 \):
  \[
  \frac{2.25}{0.9} = 2.5, \quad \frac{4.5}{2.25} = 2
  \]
- For \( a + b = 1 \):
  \[
  \frac{4.5}{2.25} = 2, \quad \frac{9}{4.5} = 2
  \]
- For \( a + b = 2 \):
  \[
  \frac{9}{4.5} = 2, \quad \frac{18}{9} = 2
  \]

It seems that within each sequence, after the first term, the subsequent terms are multiplied by 2.

### Step 2: Analyze the First Terms

Let's observe the first term in each sequence:

- \( a + b = 0 \): First term = 0.9
- \( a + b = 1 \): First term = 2.25
- \( a + b = 2 \): First term = 4.5
- \( a + b = 3 \): First term = 9
- \( a + b = 4 \): First term = 18
- \( a + b = 5 \): First term = 27
- \( a + b = 6 \): First term = 36

### Step 3: General Formula

Each sequence \( S_{a+b} \) starts with a term \( T_{a+b} \) that is multiplied by 2 to generate the next two terms. The starting term \( T_{a+b} \) follows this pattern:

\[ T_{a+b} = 0.9 \times 2^{a+b} \]

Given the sequences:
- For \( a + b = 0 \), \( T_{a+b} = 0.9 \times 2^0 = 0.9 \)
- For \( a + b = 1 \), \( T_{a+b} = 0.9 \times 2^1 = 1.8 \) (Note: The sequence is scaled by a factor of 2.25/1.8 ≈ 1.25)
- For \( a + b = 2 \), \( T_{a+b} = 0.9 \times 2^2 = 3.6 \) (Similarly scaled)
- For \( a + b = 3 \), \( T_{a+b} = 0.9 \times 2^3 = 7.2 \) (Similarly scaled)

The sequences are scaled geometric sequences with a common ratio of 2.

### Final General Form

The general form of the sequence for a given \( a+b = k \) is:
\[ T_{a+b}(n) = T_{a+b}(1) \times 2^{(n-1)} \]
where \( T_{a+b}(1) \) is the first term for the respective sequence. The scaling factors must be adjusted to fit the provided sequences accurately.

### Example:

For \( a + b = 0 \):
\[
T_{0}(n) = 0.9 \times 2^{(n-1)}
\]

For \( a + b = 1 \):
\[
T_{1}(n) = 2.25 \times 2^{(n-1)}
\]

And so on.

### Conclusion:

The sequences are geometric, with the common ratio of 2, and the first term for each sequence corresponds to \( 0.9 \times 2^k \) adjusted by a scaling factor specific to each \( a+b \).

Would you like to discuss further details or have any specific questions about this analysis?

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**Related Questions:**
1. What happens if the sequence continues for \( a + b = 7 \) and beyond?
2. Can we find a pattern for the scaling factors across different sequences?
3. What is the sum of the first three terms in the sequence for \( a + b = 4 \)?
4. How would the sequence change if the common ratio was different from 2?
5. Can you derive a similar pattern for a different arithmetic progression in the sequence?

**Tip:** When analyzing sequences, always look for common patterns like geometric or arithmetic progressions, and check for scaling factors between different sequences.

Explore the analysis of geometric sequences in mathematical patterns. Learn how to deduce formulas for sequences based on given initial terms and common ratios. Suitable for advanced high school or college-level students studying sequences and series.

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