Answer:
Explanation:
To derive a second-order equation from the sequence 1, 2, 4, 6, 10, 14, we can use the method of finite differences. The first differences are:
1, 2, 2, 4, 4
The second differences are:
1, 0, 2, 0
Since the second differences are not constant, we know that the sequence is not quadratic. However, we can still find a second-order equation that approximates the sequence.
One way to do this is to use the method of least squares to fit a curve to the data. We can assume that the curve has the form:
y = ax^2 + bx + c
where y is the nth term of the sequence and x is the index of the term (starting with x = 1 for the first term). We can then use the first six terms of the sequence to solve for a, b, and c.
Substituting x = 1, y = 1; x = 2, y = 2; and x = 3, y = 4 into the equation above gives us three equations:
a + b + c = 1
4a + 2b + c = 2
9a + 3b + c = 4
Solving these equations simultaneously gives us:
a = **1/2**
b = **-1/2**
c = **1/3**
Therefore, a second-order equation that approximates the sequence is:
y = **(1/2)x^2 - (1/2)x + (1/3)**
I hope this helps! Let me know if you have any other questions.