Answer:
x = 2n² − 4n − 3
Step-by-step explanation:
We know the sequence is quadratic, so it has the form:
x = an² + bn + c
Plug in any three points to form a system of equations:
-5 = a(1)² + b(1) + c
-3 = a(2)² + b(2) + c
3 = a(3)² + b(3) + c
-5 = a + b + c
-3 = 4a + 2b + c
3 = 9a + 3b + c
Solve the system of equations. Start by subtracting the first equation from the second and third:
-3 − (-5) = 4a + 2b + c − (a + b + c)
2 = 3a + b
3 − (-5) = 9a + 3b + c − (a + b + c)
8 = 8a + 2b
Double the top equation and subtract from the bottom:
4 = 6a + 2b
8 − 4 = 8a + 2b − (6a + 2b)
4 = 2a
a = 2
Plug into either equation to find b:
2 = 3a + b
b = -4
Finally, plug a and b into any of the first equations to find c.
-5 = a + b + c
c = -3
The nth term rule is therefore:
x = 2n² − 4n − 3
