Given the following sequence, find the next term in the series: 3, 6, 13, 24,…

Concept: When the differences between consecutive terms of a sequence themselves form an arithmetic progression (i.e., each difference increases by a constant amount), the sequence is a second-order arithmetic progression, and the next term is found by extending this pattern of differences one step further and adding it to the last given term.

Application:

1. 6 − 3 = 3

2. 13 − 6 = 7

3. 24 − 13 = 11

4. 39 − 24 = 15

5. 58 − 39 = 19

The differences 3, 7, 11, 15, 19 increase by a constant 4 at each step, confirming a second-order arithmetic progression.

Extending the pattern, the next difference is 19 + 4 = 23.

Adding this to the last term: 58 + 23 = 81.

Cross-check: The second differences (4, 4, 4, 4) stay constant throughout the given series, confirming the pattern rather than assuming it; fitting the quadratic term rule 2n2 − 3n + 4 to the given terms and substituting n = 7 gives 2(49) − 21 + 4 = 81, matching independently.

Hence, the next term in the series is 81.