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

To determine the rule for this function, let's examine the pattern in the outputs.

From the provided information, we can observe that the output values are not increasing linearly or following a simple mathematical pattern. Instead, it seems that each output value is obtained through a more complex operation.

Looking closely, it appears that the output values may be obtained by repeatedly squaring the input value and adding a constant. Let's test this pattern on the given inputs and outputs:

For input 1:
Output = (1^2) + 1 = 2
The output matches the provided value.

For input 2:
Output = (2^2) + 1 = 5
The output matches the provided value.

For input 3:
Output = (3^2) + 3 = 12
The output matches the provided value.

For input 4:
Output = (4^2) + 6 = 22
The output does not match the provided value.

Based on the given information, we cannot accurately determine the precise rule for this function without more values or additional patterns.

To find the rule for this function, we need to look for a pattern in the inputs and outputs. Let's analyze the given data:

Input 1 → Output 2
Input 2 → Output 5
Input 3 → Output 12
Input 4 → Output 34
Input 5 → Output 135
Input 6 → Output 741

From the given data, we can observe that each output is obtained by performing some mathematical operations on the corresponding input. To determine the rule, let's analyze the differences between consecutive outputs:

Output 2 → Output 5: The difference is 5 - 2 = 3
Output 5 → Output 12: The difference is 12 - 5 = 7
Output 12 → Output 34: The difference is 34 - 12 = 22
Output 34 → Output 135: The difference is 135 - 34 = 101
Output 135 → Output 741: The difference is 741 - 135 = 606

Now, let's analyze the differences between consecutive differences:

3 → 7: The difference is 7 - 3 = 4
7 → 22: The difference is 22 - 7 = 15
22 → 101: The difference is 101 - 22 = 79
101 → 606: The difference is 606 - 101 = 505

Looking at the second-order differences, we can see that the values are increasing. It suggests that the rule involves some degree of squaring or raising to a higher power. We can further analyze the third-order differences, but in this case, we see that they are not consistent.

Given the available information, it is difficult to precisely determine the rule for this function. However, based on the given data, it seems like a complex mathematical formula or algorithm is used to calculate the outputs from the inputs. It is possible that the rule involves a combination of exponentiation, multiplication, addition, or other operations. Without additional information or a clear pattern, it is challenging to deduce the exact rule for this function.