Please help, What is the nth term rule of the quadratic sequence below?
-4,1,12,29,52,81,116

If the sequence is quadratic, then the n-th term (n ≥ 1) is

We're given the first four terms,

Using the formula for the n-th term, this turns into a system of equations,

Solve the system:

• Eliminate c :

(4a + 2b + c) - (a + b + c) = 1 - (-4)     ===>     3a + b = 5

(9a + 3b + c) - (a + b + c) = 12 - (-4)     ===>     8a + 2b = 16

• Multiply the second equation by 1/2 to get 4a + b = 8, then eliminate b and solve for a :

(4a + b) - (3a + b) = 8 - 5     ===>     a = 3

• Solve for b and c :

3a + b = 9 + b = 5     ===>     b = -4

a + b + c = 3 - 4 + c = -4     ===>     c = -3

Then the rule for the n-th term is