Write out the terms, their differences and their 2nd differences.
4 13 28 49 76
9 15 21 27
6 6 6
Since it is the 2nd differences that are equal, then the nth term of the sequence is given by a 2nd degree polynomial.
tn = an^2 + bn + c, where t1 = 4, t2 = 13, t3 = 28, ...
n = 1: 4 = a + b + c -------------------- (1)
n = 2: 13 = 4a + 2b + c ----------------- (2)
n = 3: 28 = 9a + 3b + c ----------------- (3)
Subtracting (1) from (2) and (2) from (3),
9 = 3a + b ------ (4)
15 = 5a + b ----- (5)
Subtract (4) from (5).
6 = 2a
a = 3
b = 0
c = 1
Expression for the nth term is: tn = 3n^2 + 1