Find the missing number in the given series 0, 7, ....., 63, 124

A missing-number series like this is generated by a single function of the term's position n (where n = 1, 2, 3, ...). The rule is found by testing an expression against the terms that are already known; once that expression reproduces every known term exactly, the same expression gives the missing term.

1. Number the terms by position: term1 = 0, term2 = 7, term3 = missing term, term4 = 63, term5 = 124.

2. Test the expression n3 − 1 for n = 1: 13 − 1 = 1 − 1 = 0, which matches term1.

3. For n = 2: 23 − 1 = 8 − 1 = 7, which matches term2.

4. For n = 4: 43 − 1 = 64 − 1 = 63, which matches term4.

5. For n = 5: 53 − 1 = 125 − 1 = 124, which matches term5.

6. Since n3 − 1 exactly reproduces all four known terms, apply it at the missing position n = 3: 33 − 1 = 27 − 1 = 26.

Independently confirm by finite differences with 26 in place: the first differences of consecutive terms are 7, 19, 37, and 61, and the second differences of those are 12, 18, and 24 — each second difference is exactly 6 more than the previous one, the constant-third-difference behaviour expected of a cubic (n3) sequence.

The missing number is 26.