What is the rule for this function?
input 1 output 2
input 2 output 5
Input 3 output 12
Input 4 output 34
Input 5 output 135
Input 6 output 741
e^x:
2.718
7.389
20.08
54.598
148.413
403.428
so, it grows faster than e^x as x gets large
maybe e^(kx^2) can be adjusted some. Since it is not at all obvious (to me), and since there are only 6 data points, it will be e possible to fit any suitable polynomial or exponential.
For example, Newton's divided difference method fits
f(x) = 2 + 3(x-1) + 2(x-1)(x-2) + (11/6)(x-1)(x-2)(x-3) + (6625/3000)(x-1)(x-2)(x-3)(x-4) + (103/40)(x-1)(x-2)(x-3)(x-4)(x-5)
= (1/120)(309x^5 - 4370x^4 + 23835x^3 - 61330x^2 + 73476x - 31680)
To determine the rule for this function, we need to examine the pattern in the outputs based on the inputs. Let's take a look:
Input 1: Output 2
Input 2: Output 5
Input 3: Output 12
Input 4: Output 34
Input 5: Output 135
Input 6: Output 741
At first glance, it is not immediately obvious what the rule is. However, if we examine the relationship between the inputs and outputs more closely, we can identify a pattern.
Let's express the outputs as a sequence:
2, 5, 12, 34, 135, 741
If we observe the differences between consecutive terms in the sequence:
3 (5 - 2), 7 (12 - 5), 22 (34 - 12), 101 (135 - 34), 606 (741 - 135)
These differences are not yet consistent, so let's take the differences again:
4 (7 - 3), 15 (22 - 7), 79 (101 - 22), 505 (606 - 101)
Now, if we observe the differences between these second-order differences:
11 (15 - 4), 64 (79 - 15), 426 (505 - 79)
The differences of the second-order are not yet consistent either, so let's take the differences again:
53 (64 - 11), 362 (426 - 64)
Now, we have a consistent difference between the third-order differences. This suggests that the rule for the function involves a polynomial equation.
Let's assume the rule follows a pattern of the form:
f(x) = ax^3 + bx^2 + cx + d
To determine the values of a, b, c, and d, we can use the information given by the outputs corresponding to the inputs. We can create a system of equations based on this information.
Using the provided data, we can set up the following equations:
f(1) = a(1)^3 + b(1)^2 + c(1) + d = 2
f(2) = a(2)^3 + b(2)^2 + c(2) + d = 5
f(3) = a(3)^3 + b(3)^2 + c(3) + d = 12
f(4) = a(4)^3 + b(4)^2 + c(4) + d = 34
f(5) = a(5)^3 + b(5)^2 + c(5) + d = 135
f(6) = a(6)^3 + b(6)^2 + c(6) + d = 741
Simplifying these equations, we get:
a + b + c + d = 2 (equation 1)
8a + 4b + 2c + d = 5 (equation 2)
27a + 9b + 3c + d = 12 (equation 3)
64a + 16b + 4c + d = 34 (equation 4)
125a + 25b + 5c + d = 135 (equation 5)
216a + 36b + 6c + d = 741 (equation 6)
Solving this system of equations will give us the values of a, b, c, and d, and allow us to determine the rule for this function.