The sequence 0, 1, 4, 7, 8, 5, 2, 3, 6, 9, 9, 6, 3, 2, 5, 8, 7, 4, 1, 0, 0, … is given. Based on the pattern in the entire sequence, what is the next number that should come after the last 0?

Introduction / Context:
This problem presents a relatively long sequence of single digit numbers with an ellipsis indicating continuation. The values seem irregular if we only look at local changes, but the question expects us to recognize a global structure. Such sequences often hide symmetry, periodicity or reflections rather than simple arithmetic progressions.

Given Data / Assumptions:

• The sequence shown is: 0, 1, 4, 7, 8, 5, 2, 3, 6, 9, 9, 6, 3, 2, 5, 8, 7, 4, 1, 0, 0, …
• The last explicitly written term before the dots is 0.
• We assume the same pattern that produced the first 21 terms continues beyond them.
• All numbers are single digits from 0 to 9.

Concept / Approach:
A useful idea with long sequences is to look for reflection symmetry. If the sequence increases and then appears to retrace its steps in reverse order, it may be palindromic or cyclic. Another clue is the repeated pair 9, 9 in the middle which can act as a mirror point. We examine whether the first part of the sequence is mirrored in the second part and whether a cycle of fixed length repeats.

Step-by-Step Solution:
Step 1: Split the given terms into two blocks of ten numbers each. First block: 0, 1, 4, 7, 8, 5, 2, 3, 6, 9. Second block: 9, 6, 3, 2, 5, 8, 7, 4, 1, 0. Step 2: Reverse the first block: 9, 6, 3, 2, 5, 8, 7, 4, 1, 0. Step 3: Observe that the reversed first block exactly matches the second block term by term. Step 4: This shows that the first 20 terms form a symmetric pattern around the central pair 9, 9. Step 5: Note that the 21st term is 0, which is the same as the first term, suggesting that a new cycle is starting. Step 6: If the sequence is periodic with period 20, then term 21 should match term 1 (0), term 22 should match term 2 (1), and so on. Step 7: Since the 21st term is already 0, consistent with term 1, the next term (22nd) must be 1, matching term 2.

Verification / Alternative check:
We can verify by writing indices: positions 1 to 10 form the original progression; positions 11 to 20 are exactly the reverse. This demonstrates a complete symmetric cycle of length 20. Because the 21st term equals the first term, it is highly reasonable that the entire 20 term pattern is repeating, making the 22nd term equal to the second term, which is 1.

Why Other Options Are Wrong:
Choosing 0 again would break the established cycle, since after term 1 the sequence moved on to 1, not another 0. Values 4 or 7 are later positions inside the cycle and would only appear at specific indices, not immediately after restarting at 0. Therefore, only 1 keeps the pattern consistent with a repeated 20 term block.

Common Pitfalls:
Many learners focus only on local differences such as +3, +3, +1 and so on, which appear irregular and confusing. This can lead to the false conclusion that the next term cannot be determined. The key is to step back and look for global symmetric structures rather than small arithmetic changes.

Final Answer:
The sequence repeats a 20 term symmetric pattern, so after the repeated 0 at position 21, the next term must be 1.