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 or pattern for this function, we need to observe the relationship between the inputs and outputs. Let's analyze it step by step:

- When the input is 1, the output is 2.
- When the input is 2, the output is 5.
- When the input is 3, the output is 12.
- When the input is 4, the output is 34.
- When the input is 5, the output is 135.
- When the input is 6, the output is 741.

Looking at these values, it is not immediately clear what mathematical operation is happening to obtain each output. However, we can try different mathematical operations and see if any of them fit the pattern.

One approach is to look for a potential pattern by analyzing the differences between consecutive outputs. Let's calculate the differences between the outputs:

- The difference between 2 and 5 is 3.
- The difference between 5 and 12 is 7.
- The difference between 12 and 34 is 22.
- The difference between 34 and 135 is 101.
- The difference between 135 and 741 is 606.

Examining the sequence of differences, it does not seem to follow a straightforward pattern. The differences (3, 7, 22, 101, 606) do not exhibit a clear pattern themselves.

Another approach is to test different mathematical operations and see if any of them apply to the inputs and outputs. We can test common mathematical operations such as addition, subtraction, multiplication, and division.

Let's consider multiplication as a potential operation. If we multiply the input by a constant factor and add another constant, we might be able to obtain the correct outputs. Let's test this approach:

- When the input is 1, 1 * (1 * 1) + 1 = 2.
- When the input is 2, 2 * (2 * 2) + 1 = 9.
- When the input is 3, 3 * (3 * 3) + 1 = 28.
- When the input is 4, 4 * (4 * 4) + 1 = 65.
- When the input is 5, 5 * (5 * 5) + 1 = 126.
- When the input is 6, 6 * (6 * 6) + 1 = 217.

Based on these calculations, it seems that the function follows the rule of multiplying the input by the square of the input and adding 1. Therefore, the function rule is:

output = input * (input * input) + 1.