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.