General quadratic polynomial sequences can be represented by the expression:

$a_n = an^2 + bn + c$

where:

• $a$, $b$, and $c$ are constants, and
• $n$ is the term number in the sequence.

To simplify such sequences, follow these steps:

1. Identify the Sequence Formula: Given a sequence, determine the general formula by recognizing it as a quadratic sequence. The sequence is typically provided as values, such as $a_1, a_2, a_3, \dots$, where $a_n$ represents the $n$th term.

2. Calculate the First Differences: Compute the first differences by subtracting each consecutive term in the sequence: $\Delta_1 = a_{n+1} - a_n$ For quadratic sequences, the first differences will themselves form an arithmetic sequence.

3. Calculate the Second Differences: Compute the differences between consecutive first differences: $\Delta_2 = \Delta_{1,n+1} - \Delta_{1,n}$ For quadratic sequences, the second differences will be constant.

4. Determine the Coefficients $a$, $b$, and $c$:

   • The constant second difference is equal to $2a$.
   • Use the system of equations derived from substituting small values of $n$ (like 1, 2, 3) into the general formula $a_n = an^2 + bn + c$ to solve for $a$, $b$, and $c$.

5. Write the Simplified Sequence Formula: Substitute the calculated coefficients back into the general formula to obtain the simplified sequence formula.

Example

Given a sequence: $2, 6, 12, 20, \dots$

1. First differences: $6 - 2 = 4$, $12 - 6 = 6$, $20 - 12 = 8$
2. Second differences: $6 - 4 = 2$, $8 - 6 = 2$ (constant)

Since the second difference is constant and equal to 2, we have $2a = 2 \Rightarrow a = 1$.

Using the first term $a_1 = 2$: $a_n = n^2 + bn + c$ Substitute $n = 1$: $1^2 + b(1) + c = 2 \Rightarrow 1 + b + c = 2$ Substitute $n = 2$ to find $b$ and $c$.

Would you like to see further details of this process?

Relative Questions:

1. How do you determine the first differences in a sequence?
2. What indicates that a sequence is quadratic based on differences?
3. How are the coefficients $a$, $b$, and $c$ calculated?
4. Can you describe the significance of constant second differences?
5. What is the method to verify the simplified sequence formula?

Tip:

Remember that for quadratic sequences, the second differences are always constant and directly relate to the coefficient $a$ in the general form $an^2 + bn + c$.