To determine the eighth number in the sequence using the method of successive differences, follow these steps:

1. Write down the sequence:
   $3, 13, 29, 55, 95, 153, 233$

2. Calculate the first differences:
   Subtract each term from the next: $13 - 3 = 10$ $29 - 13 = 16$ $55 - 29 = 26$ $95 - 55 = 40$ $153 - 95 = 58$ $233 - 153 = 80$ So, the first differences are: $10, 16, 26, 40, 58, 80$

3. Calculate the second differences:
   Subtract each first difference from the next: $16 - 10 = 6$ $26 - 16 = 10$ $40 - 26 = 14$ $58 - 40 = 18$ $80 - 58 = 22$ So, the second differences are: $6, 10, 14, 18, 22$

4. Calculate the third differences:
   Subtract each second difference from the next: $10 - 6 = 4$ $14 - 10 = 4$ $18 - 14 = 4$ $22 - 18 = 4$ So, the third differences are constant: $4, 4, 4, 4$

Since the third differences are constant, the sequence follows a cubic pattern.

5. Extend the third differences:
   The third differences remain constant, so we extend the differences table upward to find the eighth term.

6. Extend the second differences:
   Add 4 to the last second difference: $22 + 4 = 26$ So, the next second difference is 26.

7. Extend the first differences:
   Add 26 to the last first difference: $80 + 26 = 106$ So, the next first difference is 106.

8. Extend the sequence:
   Add 106 to the last term of the sequence: $233 + 106 = 339$

Thus, the eighth number in the sequence is 339.

Would you like more details or have any questions? Here are some related questions to expand on:

1. What is the general formula for sequences with constant third differences?
2. Can the method of successive differences be applied to other types of patterns?
3. How does polynomial interpolation relate to successive differences?
4. What are some applications of cubic sequences in real-world problems?
5. How do we verify the correctness of a sequence once the eighth term is found?

Tip: Successive differences are especially useful for identifying polynomial patterns in number sequences.