Question

explain THE STEPS TO CONstruct an appROXIMATE polynomial expression to represent a sequence with constant 2nd differences

Answer 1

To construct an approximate polynomial expression for a sequence with constant second differences, you generally want to find a quadratic polynomial (since constant second differences indicate that the sequence can be represented by a polynomial of degree 2). Here are the steps you can follow:

### Step 1: Determine the Sequence

Start with the sequence you want to analyze. For example, you may have:
\[ a_0, a_1, a_2, a_3, \ldots \]

### Step 2: Calculate the First Differences

Calculate the first differences of the sequence. The first difference is found by subtracting each term from the next term:
\[
\Delta_1 = a_1 - a_0, \quad \Delta_2 = a_2 - a_1, \quad \Delta_3 = a_3 - a_2, \ldots
\]
This gives you a new sequence of first differences.

### Step 3: Calculate the Second Differences

Next, calculate the second differences by taking the differences of the first differences:
\[
\Delta^2_1 = \Delta_2 - \Delta_1, \quad \Delta^2_2 = \Delta_3 - \Delta_2, \ldots
\]
If the second differences are constant, denote this constant as \(c\).

### Step 4: Formulate the General Quadratic Polynomial

Since the second differences are constant, you can express the \(n\)-th term \(a_n\) of the sequence as a quadratic polynomial of the form:
\[
a_n = An^2 + Bn + C
\]
where \(A\), \(B\), and \(C\) are coefficients that you will determine.

### Step 5: Set Up a System of Equations

To find the coefficients \(A\), \(B\), and \(C\), you can use a set of known values from the sequence. Usually, the three consecutive terms \(a_0\), \(a_1\), and \(a_2\) are used to set up the equations.

1. \(a_0 = A(0^2) + B(0) + C \Rightarrow C = a_0\)
2. \(a_1 = A(1^2) + B(1) + C \Rightarrow a_1 = A + B + a_0\)
3. \(a_2 = A(2^2) + B(2) + C \Rightarrow a_2 = 4A + 2B + a_0\)

### Step 6: Solve the System of Equations

Now, substitute \(C\) back into the equations derived for \(a_1\) and \(a_2\) to get a system of equations in terms of \(A\) and \(B\):
1. \(A + B = a_1 - a_0\)
2. \(4A + 2B = a_2 - a_0\)

You can solve this system to find the values of \(A\) and \(B\).

### Step 7: Write the Final Expression

Once you have determined the values of \(A\), \(B\), and \(C\), substitute them back into the quadratic polynomial:
\[
a_n = An^2 + Bn + C
\]

### Step 8: Verify the Expression

Finally, check the polynomial by substituting different values of \(n\) into the polynomial to see if it matches the original sequence. It’s a good idea to confirm that the polynomial generates the expected values of the sequence.

By following these steps, you can construct an approximate polynomial expression for a given sequence characterized by constant second differences.