The next term in the series −1, 5, 15, 29, ?, ... is :

Concept — In a number series the first differences are the gaps between consecutive terms. When those gaps are unequal, take the differences of the gaps themselves; these are the second differences. A constant second difference means the series is quadratic in its position n, so its general term has the form T(n) = an2 + bn + c, and each new gap equals the previous gap plus that constant.

Application — Work through the listed terms −1, 5, 15, 29 in order.

1. First differences: 5 − (−1) = 6, then 15 − 5 = 10, then 29 − 15 = 14.

2. Second differences: 10 − 6 = 4 and 14 − 10 = 4. The second difference is constant, so the series is quadratic in n.

3. The next first difference therefore continues the same way: 14 + 4 = 18.

4. Add that gap to the last listed term: 29 + 18 = 47.

Cross-check — A constant second difference of 4 fixes the leading coefficient at a = 4 ÷ 2 = 2. Substituting T(1) = −1 and T(2) = 5 into T(n) = an2 + bn + c then gives b = 0 and c = −3, so T(n) = 2n2 − 3. Every listed term matches:

• T(1) = 2(1)2 − 3 = −1

• T(2) = 2(2)2 − 3 = 5

• T(3) = 2(3)2 − 3 = 15

• T(4) = 2(4)2 − 3 = 29

• T(5) = 2(5)2 − 3 = 47

Contrast — Reaching 36, 59 or 63 from 29 would need jumps of 7, 30 or 34; none of those keeps the second difference at 4, and none of those values equals 2n2 − 3 at n = 5.

Result — the term that follows 29 in this series is 47.