Since we know it's quadratic, the n-th term will follow the pattern
[tex]x_n = an^2 + bn + c[/tex]
for some unknown coefficients a, b, and c.
Given that [tex]x_1=-1[/tex], [tex]x_2=2[/tex], and [tex]x_3=7[/tex], we have the following conditions on these coefficients:
[tex]\begin{cases} a + b + c = -1 \\ 4a + 2b + c = 2 \\ 9a + 3b + c = 7 \end{cases}[/tex]
Solve this system to get a = 1, b = 0, and c = -2. Then
[tex]\boxed{x_n = n^2 - 2}[/tex]
To solve the system, use elimination.
[tex](4a + 2b + c) - (a + b + c) = 2 - (-1) \implies 3a + b = 3[/tex]
[tex](9a + 3b + c) - (a + b + c) = 7 - (-1) \implies 8a + 2b = 8 \implies 4a + b = 4[/tex]
[tex](4a + b) - (3a + b) = 4 - 3 \implies a = 1 \implies b = 0 \implies c = -2[/tex]
