 3, 13, 29, 55, 93, 153, 233 Use the method of successive differences to determine the next number in the sequence.


To determine the next number in the sequence using the method of successive differences, let's follow these steps:

1. **Find the first differences** between consecutive terms in the sequence:

   - \( 13 - 3 = 10 \)
   - \( 29 - 13 = 16 \)
   - \( 55 - 29 = 26 \)
   - \( 93 - 55 = 38 \)
   - \( 153 - 93 = 60 \)
   - \( 233 - 153 = 80 \)

   So, the first differences are: \(10, 16, 26, 38, 60, 80\).

2. **Find the second differences** between the first differences:

   - \( 16 - 10 = 6 \)
   - \( 26 - 16 = 10 \)
   - \( 38 - 26 = 12 \)
   - \( 60 - 38 = 22 \)
   - \( 80 - 60 = 20 \)

   So, the second differences are: \(6, 10, 12, 22, 20\).

3. **Find the third differences** between the second differences:

   - \( 10 - 6 = 4 \)
   - \( 12 - 10 = 2 \)
   - \( 22 - 12 = 10 \)
   - \( 20 - 22 = -2 \)

   So, the third differences are: \(4, 2, 10, -2\).

The differences are not consistent, which indicates that this sequence might not fit perfectly into a simple polynomial pattern, and it might involve a more complex rule. However, since there's an identifiable pattern within each level of differences, we can use these to extrapolate the next term.

**Extrapolating the next term:**

Looking at the first differences: \(10, 16, 26, 38, 60, 80\), we could continue this sequence by predicting the next difference. By examining these values, it's evident that this sequence roughly follows a pattern that increases irregularly. Assuming the next first difference might follow the incremental growth pattern:

- Potential next difference in the first differences: \(80 + 20 = 100\).

**Adding this to the last term in the original sequence:**

- \(233 + 100 = 333\).

So, the next term in the sequence is **333**.

Would you like to know more details or have questions? 

Here are some related questions:

1. How do you identify patterns in sequences with irregular growth?
2. Can all sequences be solved using successive differences?
3. How would you find the next term in a sequence if the differences do not stabilize?
4. What other methods can be used to find the next term in complex sequences?
5. How can polynomial fitting help in predicting terms in sequences?

**Tip:** Successive differences work well for sequences that can be expressed as polynomials. If the differences don't stabilize, consider exploring more advanced or alternative methods, such as polynomial fitting or recursive relations.

3, 13, 29, 55, 93, 153, 233 Use the method of successive differences to determine the next number in the sequence.

Learn how to determine the next number in the sequence 3, 13, 29, 55, 93, 153, 233 using the method of successive differences. Follow a step-by-step solution that covers first, second, and third differences and explains how to predict the next term in the sequence. Suitable for Grades 10-12.

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