Write an expression for the apparent nth term (an) of the sequence. (Assume that n begins with 1.)
−8, −5, 0, 7, 16, . . .
an=
sequence: -8 -5 0 7 16
finding the differences: 3 5 7 9
finding those differences: 2 2 2
since the second set of difference is a constant, this will be a quadratic expression: ax2 + bx + c
at (1, -8) a(1)2 + b(1) + c = -8 ---> a + b + c = -8
at (2, -5) a(2)2 + b(2) + c = -5 ---> 4a + 2b + c = -5
at (3, 0) a(3)2 + b(3) + c = 0 ---> 9a + 3b + c = 0
Solving for a, b, and c, we get a = 1 b = 0 and c = -9
Therefore, the expression for the nth term is: n2 - 9