a+b=7: 45, 90, 180. Please verify and solve

Let's verify and understand how the sequence given for \( a + b = 7 \) fits into the existing pattern and deduce if it follows the established rule.

### Given Sequence for \( a + b = 7 \):
- Sequence: \( 45, 90, 180 \)

### Step 1: Check the Ratio Between Consecutive Terms

Let's check if the sequence maintains the same ratio between consecutive terms:
- \( \frac{90}{45} = 2 \)
- \( \frac{180}{90} = 2 \)

The ratio between each consecutive term is indeed 2, consistent with the previous sequences. This confirms that the sequence is a geometric sequence with a common ratio of 2.

### Step 2: Verify the First Term with the Established Pattern

The first term in each sequence can be expressed as \( 0.9 \times 2^{a+b} \).

For \( a + b = 7 \), the first term should be:
\[
T_{7}(1) = 0.9 \times 2^7 = 0.9 \times 128 = 115.2
\]
However, the first term given is 45, which does not match 115.2.

### Step 3: Adjusting for Scaling Factor

Let's see if the sequence can be related to a scaling of the sequence by some factor. 

If the first term is 45, we can determine the factor by comparing it to the expected 115.2:
\[
\text{Scaling Factor} = \frac{45}{115.2} = \frac{45}{115.2} \approx 0.39
\]

This scaling factor seems arbitrary, which suggests that the sequence may not follow the exact formula derived earlier or that an adjustment has been made.

### Step 4: Confirm the Sequence Consistency

Since the common ratio is 2 and the sequence is consistent:
- If we start at 45:
  \[
  45 \times 2 = 90
  \]
  \[
  90 \times 2 = 180
  \]
This means the sequence \( 45, 90, 180 \) is consistent and correct based on the given terms.

### Conclusion:

The sequence \( 45, 90, 180 \) for \( a + b = 7 \) is a geometric sequence with a common ratio of 2. However, the first term does not fit perfectly into the earlier deduced pattern of \( 0.9 \times 2^{a+b} \), indicating that a scaling factor might be involved, or the sequence has been adjusted from the original pattern.

Would you like to delve deeper into understanding this discrepancy, or perhaps explore another aspect of the sequence?

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**Related Questions:**
1. What would the sequence be if the common ratio was 3 instead of 2?
2. Can we determine the scaling factor for other sequences if the pattern is broken?
3. How would you predict the next term in this sequence for \( a + b = 8 \)?
4. Is there an alternative method to generate these sequences without a geometric progression?
5. Can you derive a continuous function that represents all these sequences as \( a + b \) increases?

**Tip:** When a sequence doesn't perfectly follow the expected pattern, check for possible scaling factors or modifications. Small deviations often indicate an underlying rule that may be slightly adjusted.

Explore the geometric sequence defined by the equation a+b=7 with the sequence 45, 90, 180. Learn about the pattern's consistency and understand the role of scaling factors in such sequences. Suitable for high school students studying geometric patterns in sequences.

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