28, 42, 75, 135, (A), (B), 442, 630 Find A and B

To find \( A \) and \( B \) in the sequence 28, 42, 75, 135, (A), (B), 442, 630, we need to observe the pattern or rule governing the sequence. Let's analyze the differences between the consecutive terms:

1. \( 42 - 28 = 14 \)
2. \( 75 - 42 = 33 \)
3. \( 135 - 75 = 60 \)
4. \( A - 135 = ? \)
5. \( B - A = ? \)
6. \( 442 - B = ? \)
7. \( 630 - 442 = 188 \)

Now, let's look at how the differences change:
- \( 14 \to 33 \to 60 \)

The differences between consecutive terms seem to increase. Let's look at the difference between the differences:
- \( 33 - 14 = 19 \)
- \( 60 - 33 = 27 \)

The second-order difference is increasing by 8, suggesting the differences between consecutive terms might continue to increase by 8.

Thus, the next difference should be:
- \( 60 + 35 = 95 \) (since 27 + 8 = 35)

So, we can now calculate \( A \):
- \( A = 135 + 95 = 230 \)

Next, the difference between \( A \) and \( B \) should increase by another 8:
- \( 95 + 43 = 138 \)

Thus, \( B = 230 + 138 = 368 \).

Let's check the next difference:
- \( 442 - 368 = 74 \)

This checks out, since the next difference (74) follows the increasing pattern.

So, \( A = 230 \) and \( B = 368 \).

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Would you like a detailed explanation or have any questions?

Here are 5 related questions to expand on the topic:
1. How can we determine patterns in more complex sequences?
2. What is the significance of second-order differences in sequences?
3. Can this approach be used for sequences that don’t follow arithmetic progressions?
4. What are other methods to find missing terms in a sequence?
5. How would this pattern apply if we had fractional numbers in the sequence?

**Tip:** When analyzing sequences, checking second or third-order differences can reveal hidden patterns not immediately obvious from the first set of differences.

Discover how to find missing terms A and B in the sequence 28, 42, 75, 135, (A), (B), 442, 630 by analyzing the second-order differences. This solution involves pattern recognition and arithmetic progression, perfect for students in Grades 8-10.

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