**Contents:**

- What is a Quadratic Sequence?
- How to Verify a Sequence is Quadratic
- Finding the nth Number in a Quadratic Sequence
- Finding the General Form

## What is a Quadratic Sequence?

A **quadratic sequence** has the general form

**T(n) = an ^{2} + bn + c.**

Where a, b, and c are constants.

For example, all of the following are quadratic sequences:

- 3, 6, 10, 15, 21,…
- 7, 17, 31, 49, 71,…
- 31, 30, 27, 22, 15,…

Quadratic sequences are related to squared numbers because each sequence includes a squared number an^{2}. For example, the formula n^{2} + 1 gives the sequence: 2, 5, 10, 17, 26, …. That sequence was obtained by plugging in the numbers 1, 2, 3, … into the formula an^{2}:

- 1
^{2}+ 1 = 2 - 2
^{2}+ 1 = 5 - 3
^{2}+ 1 = 10 - 4
^{2}+ 1 = 17 - 5
^{2}+ 1 = 26

More formally, we say that a quadratic sequence has its *n*th terms given by a quadratic function **T(n) = an ^{2} + bn + c.**

## How to Verify a Sequence is Quadratic

A *sequence* is an ordered list of numbers and each number in the sequence is called a *term*. Each term in a quadratic sequence is related by the same *common second difference*. It’s called a common *second* difference (or second order difference) because you have to find the difference between each term twice. **Second order differences in quadratic sequence are always constant.** So, to verify that a particular sequence is quadratic, find the common second difference and verify that those differences are constant.

**Example question:** Verify the sequence

3, 12, 25, 42 63, …

is quadratic:

Step 1: Subtract each term from the next (i.e. subtract the second term from the first, the third from the second, and so on):

- 12 – 3 = 9,
- 25 – 12 = 13,
- 42 – 25 = 17,
- 63 – 42 = 21.

Step 2: Find the difference between the terms you found in Step 1 (9, 13, 17, 21,…)

- 13 – 9 = 4,
- 17 – 13 = 4,
- 21 – 17 = 4.

The common second difference is a constant (4), so 3, 12, 25, 42 63, … is a quadratic sequence.

## Finding the nth Number in a Quadratic Sequence

To find the *n*th number, plug that number into a given formula. For example, to find the 10th number in the sequence n^{2} + 1:

- 10
^{2}+ 1 = 101.

## Finding the General Form

The general form of a quadratic sequence follows **T(n) = an ^{2} + bn + c.**. So, given a sequence of numbers, your goal is to identify a, b, and c (the coefficients).

**Example question:** Find the general form of the quadratic sequence 6, 11, 18, 27, 38, 51:

Step 1: Find the first coefficient (a):

- Find the common second difference:
- 11 – 6 = 5,
- 18 – 11 = 7,

Giving a common second difference of 2 (because 7 – 5 = 2). We only needed to calculate the first couple of terms here because we don’t need to verify it’s quadratic, only calculate the second difference for the formula.

- Use algebra to solve the formula 2a = 2nd difference. Our second difference here is 2, so (plugging that into the formula), we get:
- 2a = 2
- a = 2/2 = 1

Out first coefficient is **1**.

Step 2: Find the second coefficient (b) using the formula:

3a + b = difference between the first and second terms.

The difference between terms 1 and 2 is:

- 11 – 6 =
**5**,

Putting that into the formula (along with “a” from Step 1) we get:

- 3(1) + b = 5
- 3 + b = 5
- b = 2

Our second coefficient (b) is 2.

Step 3: Find the third coefficient (c) using the formula a + b + c = first term.

For this particular sequence, the first term is **6**. Plugging that into the formula (along with “a” from Step 1 and “b” from Step 2):

- 1 + 2 + c = 6,
- 3 + c = 6,
- c = 3.

Our third coefficient is 3.

Step 4: Place your answers from Steps 1 to 3 into the formula T(n) = an^{2} + bn + c:

T(n) = 1n^{2} + 2n + 3 T(n) = n^{2} + 2n + 3.

If you have a calculator that has a regression feature, this video is a great hack:

## References

Surowski, D. IB Mathematics HL — Year 1. Unit 3: Sequence, Series, Binomial Theorem and Counting Arguments. Retrieved January 17, 2020 from: https://www.math.ksu.edu/~dbski/IBY1/unit3_homework.pdf

Yee, L. Sequencing Math DNA: Differences, Nth Terms, and Algebraic Sequences.

Image: N. Mori | Wikimedia Commons (Public Domain).