Question

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

Answer 1

Answer: 123

Step-by-step explanation: You start by adding 2. After that, you add 4 to two, which is six, and add six. After six you add ten, and so on. So you would eventually get to 93, the eighth term, where you would then add 30 to get 123.

Answer 2

x = 2n² − 4n − 3

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