What will be the next number in the sequence? (-1, 4, 11, 20, 31, 44, ?)

Key insight: the differences between consecutive terms are odd numbers that increase by 2.

Differences: 5, 7, 9, 11, 13, 15

• -1 + 5 = 4

• 4 + 7 = 11

• 11 + 9 = 20

• 20 + 11 = 31

• 31 + 13 = 44

• 44 + 15 = 59

Answer: 59

Alternative derivation:

Fit a quadratic formula t_n = n^2 + 2n - 4 (with n = 1 giving -1). For n = 7, t_7 = 49 + 14 - 4 = 59.