What is the nth term rule of the quadratic sequence -5,-3,3,13,27,45,67

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

180

step-by-step explanation:

f(x) has value 150 at x= 0

and 0 at x= 3

so d(f(x))/dt = slope = -150/3 = -50