Write a rule for the pattern.
-3,-9,-14,-47,-141,.....
(0,0)(2,10)(4,20)(6,30)
To find the rule for the given pattern, let's look for the differences between consecutive terms:
1st difference: -9 - (-3) = -6
2nd difference: -14 - (-9) = -5
3rd difference: -47 - (-14) = -33
4th difference: -141 - (-47) = -94
From the 1st, 2nd, 3rd, and 4th differences, we can see that the differences are not constant. This suggests that the pattern may involve a quadratic relationship.
To determine the quadratic relationship, we can start by assuming a rule of the form:
An^2 + Bn + C
where n represents the position of each term in the pattern, and A, B, and C are constants.
Now, let's substitute the values of n and the corresponding terms into this equation to form a system of equations:
For the first term (n = 1):
A(1)^2 + B(1) + C = -3
For the second term (n = 2):
A(2)^2 + B(2) + C = -9
For the third term (n = 3):
A(3)^2 + B(3) + C = -14
Simplifying these equations, we have:
A + B + C = -3
4A + 2B + C = -9
9A + 3B + C = -14
Solving this system of equations will give us the values of A, B, and C, which will provide the rule for the pattern.