What is the algebraic rule for x = 1, 2, 3, 4, ... and y = 3, 9, 18, 30, ...
y=(3/2)(x^2+x)
To find the algebraic rule for a sequence of numbers, we need to observe the pattern and try to find a relationship between the terms. In this case, we can notice that the second term (9) is 3 times the first term (3), the third term (18) is 2 times the second term (9), and the fourth term (30) is 1.67 times the third term (18).
This suggests that the pattern could be the terms are increasing by a multiple of 3 each time. Let's test this hypothesis by calculating the differences between consecutive terms:
1st difference: 9 - 3 = 6
2nd difference: 18 - 9 = 9
3rd difference: 30 - 18 = 12
The differences are not constant, indicating that the pattern may not be a simple linear relationship. However, if we take the ratios of consecutive differences, we observe:
2nd difference / 1st difference = 9 / 6 = 1.5
3rd difference / 2nd difference = 12 / 9 = 1.33
The ratios are approximately constant. This suggests that the pattern may be described by a quadratic equation.
To find the quadratic equation, let's assume that the nth term of the sequence can be represented as:
y = ax^2 + bx + c
We can now substitute the given values for x and y to form a system of equations:
When x = 1, y = 3:
3 = a(1)^2 + b(1) + c ---> a + b + c = 3 ----(1)
When x = 2, y = 9:
9 = a(2)^2 + b(2) + c ---> 4a + 2b + c = 9 ----(2)
When x = 3, y = 18:
18 = a(3)^2 + b(3) + c ---> 9a + 3b + c = 18 ----(3)
We now have a system of equations that we can solve simultaneously to find the values of a, b, and c. By solving this system of equations, you will obtain the values for a, b, and c.