If the inputs are: 1, 2, 3, 4, 5 and my outputs are: 1, 5, 11, 19, 29, what is the rule?
Starting with 4, each succeeding output is increased by the next even number.
I'm sorry. I'm only in 6th grade though, so I don't know what you mean when you say that.
1 + 4 = 5
5 + 6 = 11
11 + 8 = 19
10 + 10 = 29
1^2 + 0 = 1
2^2 + 1 = 5
3^2 + 2 = 11
4^2 + 3 = 19
5^2 + 4 = 29
To find the rule that governs the relationship between the given inputs and outputs, we can look for patterns and make observations. In this case, let's examine the differences between consecutive inputs and outputs.
Input: 1 → 2 → 3 → 4 → 5
Output: 1 → 5 → 11 → 19 → 29
Calculating the differences:
Difference between inputs: 2 - 1 = 1; 3 - 2 = 1; 4 - 3 = 1; 5 - 4 = 1
Difference between outputs: 5 - 1 = 4; 11 - 5 = 6; 19 - 11 = 8; 29 - 19 = 10
From this observation, we can observe that the differences between consecutive outputs increase by 2 each time. This suggests that the rule for generating the outputs is likely a quadratic relationship.
To confirm this, we can check if the differences between the outputs form a linear pattern. If they do, this means the rule is likely quadratic.
Checking the differences between the differences:
Difference between differences: 6 - 4 = 2; 8 - 6 = 2; 10 - 8 = 2
Since the differences between the differences are constant at 2, it confirms that the rule is quadratic.
To determine the quadratic rule, we need to find a quadratic equation that fits the given input-output relationship.
Let's assume the quadratic rule can be expressed as:
Output = ax^2 + bx + c
Given the inputs and outputs:
Input: 1 → 2 → 3 → 4 → 5
Output: 1 → 5 → 11 → 19 → 29
Substituting the inputs and outputs into the quadratic equation:
1 = a(1)^2 + b(1) + c
5 = a(2)^2 + b(2) + c
11 = a(3)^2 + b(3) + c
19 = a(4)^2 + b(4) + c
29 = a(5)^2 + b(5) + c
Simplifying the equations:
a + b + c = 1 (Equation 1)
4a + 2b + c = 5 (Equation 2)
9a + 3b + c = 11 (Equation 3)
16a + 4b + c = 19 (Equation 4)
25a + 5b + c = 29 (Equation 5)
By solving this system of linear equations, we can find the values of a, b, and c, which will give us the full quadratic rule.
After calculating this system of equations, we find that the values are:
a = 2
b = -3
c = 2
Therefore, the quadratic rule for the given inputs and outputs is:
Output = 2x^2 - 3x + 2.