Check the forward differences of the sequence.
If , then let
be the sequence of first-order differences of
. That is, for n ≥ 1,
so that .
Let be the sequence of differences of
,
and we see that this is a constant sequence, . In other words,
is an arithmetic sequence with common difference between terms of 2. That is,
and we can solve for in terms of
:
and so on down to
We solve for in the same way.
Then
and so on down to
Answer:
7 - (-14)
7 + 14
21
Answer:
Step-by-step explanation:
Recall that one needs just three points on the plane to determine the form of the quadratic function that goes through them, because a quadratic function is defined by three parameters. We therefore can pick just three simple ones that facilitate our calculations.
Realize that we need to find the parameters and
that gives as a function of the form:
that satisfies the values given in the table.
The simplest point to start with is the point (0,-6) which means that when x is 0 (zero), the value of y must be "-6". Placing such values in the general form, renders what the value of the parameter "c" should be:
So parameter must be "-6".
Now let's use other simple values of "x" that facilitate our calculations, and include the value we just found, to reduce the number of unknowns:
point (1, 2):
Point (-1, -20):
Now, we can add these two equations term by term as they are, in order to get rid of the unknown "b":
So we just found the value for parameter "" = -3
Now we can use it as well as c = -6 in one of the equations we reduced above, in order to find the parameter (b) that we are missing:
So now we have the three parameters we need to define the quadratic function:
You can also check that the other points given in the table (2, 4) a,d (3, 0) are indeed points that belong to this quadratic function.