6, 11, 21, 36, 56, ? What will replace the question mark (?) in the series?

When the first differences between consecutive terms of a sequence themselves increase by a constant amount, the sequence is called a second-order (quadratic) sequence. The next term is found by identifying the pattern in the successive differences, extending that pattern by one more step, and adding the result to the last given term.

1. Given sequence: 6, 11, 21, 36, 56, ?

2. First differences between consecutive terms: 11 - 6 = 5; 21 - 11 = 10; 36 - 21 = 15; 56 - 36 = 20

3. The differences themselves increase by a constant amount at each step: 5, 10, 15, 20 - that is, 5×1, 5×2, 5×3, 5×4

4. Extending this pattern one more step, the fifth difference = 5 × 5 = 25

5. Add this difference to the last given term: 56 + 25 = 81

Cross-check: since the successive differences increase by a constant amount, the sequence follows a quadratic rule of the form T(n) = an2 + bn + c. Fitting this to the given terms gives T(n) = 2.5n2 - 2.5n + 6, and evaluating at n = 6 also gives T(6) = 81, confirming the result obtained from the difference pattern.