Answer:
6n^2 - 5n + 3
Explanation:
To find the nth term rule for the given quadratic sequence, we need to determine the pattern or relationship between the terms. Looking at the sequence:
4, 22, 52, 52, 94, 148,...
We can observe that the first term is 4, the second term is 22 (which is 18 more than the first term), and the third term is 52 (which is 30 more than the second term).
To find the nth term rule, we will first find the differences between consecutive terms:
1st difference: 18, 30, 0, 42, ...
We notice that the 2nd difference (the differences between the differences) is constant, which suggests that the sequence follows a quadratic pattern.
2nd difference: 12, -30, 42, ...
Now, to find the nth term rule, we can use the general form of a quadratic sequence:
an = dn^2 + en + f
By substituting the values of the terms into the equation, we can find the coefficients d, e, and f.
Let's use the first three terms to form three equations:
For the 1st term (n = 1):
4 = d(1)^2 + e(1) + f
4 = d + e + f ...(1)
For the 2nd term (n = 2):
22 = d(2)^2 + e(2) + f
22 = 4d + 2e + f ...(2)
For the 3rd term (n = 3):
52 = d(3)^2 + e(3) + f
52 = 9d + 3e + f ...(3)
Solving these three equations simultaneously will give us the values of d, e, and f.
Subtracting equation (1) from equation (2):
18 = 3d + e ...(4)
Subtracting equation (1) from equation (3):
48 = 8d + 2e ...(5)
Now, subtracting equation (4) from equation (5):
30 = 5d
d = 6
Substituting the value of d into equation (4):
18 = 3(6) + e
e = -5
Substituting the value of d into equation (1):
4 = 6 + (-5) + f
f = 3
Therefore, the nth term rule for this quadratic sequence is:
an = 6n^2 - 5n + 3