Find the number that can replace the question mark (?) in the series given…

Concept — An interleaved number series is generated by a single parameter that advances in fixed steps, with each value of that parameter producing two consecutive terms through two fixed formulas. Recovering the step and the two formulas turns the series into a rule that can be evaluated at any position, instead of a difference pattern that has to be guessed.

Application

1. Separate the terms by position. The 1st, 3rd, 5th and 7th terms are 3, 24, 63 and the missing entry; the 2nd, 4th, 6th and 8th terms are 6, 30, 72 and 132.

2. Write each odd-position term as one less than a perfect square: 3 = 22 − 1, 24 = 52 − 1, 63 = 82 − 1. The bases 2, 5, 8 increase by 3 each time, so the next base is 11.

3. Test the same bases on the even-position terms: 6 = 22 + 2, 30 = 52 + 5, 72 = 82 + 8, 132 = 112 + 11. Every given even-position term is reproduced, which confirms that the parameter runs 2, 5, 8, 11.

4. Evaluate the odd-position formula at the base 11: 112 − 1 = 121 − 1 = 120.

Cross-check

• The odd-position terms 3, 24, 63 rise by 21 and then by 39, a constant second difference of 18; the next rise is therefore 57, and 63 + 57 = 120.

• Within each pair the second term exceeds the first by 3, 6 and 9 (6 − 3, 30 − 24, 72 − 63), so the fourth pair's gap is 12 and 132 − 12 = 120.

Result — The number that replaces the question mark is 120.