How do you sum a quadratic sequence?

To sum a quadratic sequence, you can use a formula derived from the sequence's properties. Here's how it works:

Understanding Quadratic Sequences

A quadratic sequence has a constant second difference. This means the differences between consecutive terms form an arithmetic sequence. Let's break down the key components:

• First term (a): The first number in the sequence.
• First difference (d): The difference between the second and first term, and so on. These differences are not constant in a quadratic sequence.
• Constant difference (c): This is the difference between the differences (the second differences), and is constant in a quadratic sequence.

Calculating the Sum

The reference mentions that by using the values of the first term (a), first difference (d), and constant difference (c), you can derive a formula in the form an² + bn + c, where n is the number of terms. This formula directly gives the sum of the first n terms of a quadratic sequence. Note that these a, b, and c are different than the first term, first difference, and constant second difference.

To use this in practice, you'll need to find the a, b, and c parameters that defines a formula, which can then be used to calculate the sum of the first n terms. To obtain these parameters, you would typically apply several known formulas in terms of first term, first differences, and the second difference.

Example:

Let's say you have a quadratic sequence where:

• a = 1 (first term of the sequence)
• The first differences are 3, 5, 7...
• The constant second difference is 2.

Following the method outlined above and in the reference, you’d end up with a sum formula like Sn = n² + 0n + 0 or simply Sn = n². Therefore to get the sum of the first 4 terms (1+4+9+16) using the Sn formula:
S4= 4²
S4= 16
The value of n is the number of terms you are summing.

Steps for Finding the Sum Formula

1. Identify the terms: List out the first few terms of the sequence.
2. Calculate the differences: Determine the first differences between consecutive terms and then the second differences.
3. Confirm quadratic nature: Verify if the second difference is constant. If so, it's a quadratic sequence.
4. Derive a sum formula: Using the reference information (or known methods for deriving the formula) calculate the specific coefficients in a quadratic expression in the form an² + bn + c which allows you to find sum, Sn.
5. Calculate the sum: Once you have the sum formula, use it to calculate the sum of any number of terms, n.

Note: This formula and its derivation is different than the formula for the nth term.