What is the rule for this function? Input 1 and output 1, input 2 and output 3, input 3 and output 6, input 4 and output 10
the sequence 1,3,6,10,... should be familiar. If it's not, remember it; it will show up frequently.
y = x(x+1)/2
To find the pattern in this function, we can look at the differences between successive outputs.
For example:
- The difference between output 1 and output 3 is 2.
- The difference between output 3 and output 6 is 3.
- The difference between output 6 and output 10 is 4.
As we can see, the differences are increasing by 1 each time. This suggests that the function is quadratic, meaning it is likely to be of the form f(x) = ax^2 + bx + c, where a, b, and c are constants.
To determine the values of a, b, and c, we can substitute the given inputs and outputs into the equation and solve the resulting system of equations.
Using the given inputs and outputs:
- When input is 1 (x = 1), the output is 1 (f(1) = 1).
- When input is 2 (x = 2), the output is 3 (f(2) = 3).
- When input is 3 (x = 3), the output is 6 (f(3) = 6).
- When input is 4 (x = 4), the output is 10 (f(4) = 10).
Substituting these values into the equation, we get the following system of equations:
a + b + c = 1
4a + 2b + c = 3
9a + 3b + c = 6
16a + 4b + c = 10
Solving this system of equations will yield the values of a, b, and c, which will define the rule for this function.