Skip to content# Quadratic Sequence: GCSE Questions, Examples with and Worksheet

## What is a Quadratic Sequence?

### Practice Questions

**Solution:**

## Formula for Quadratic Sequence

### Practice Questions

**Solution:**

## Finding the nth term of a Quadratic Sequence

## Exam Tips...

### Practice Questions

**Solution:**

## Worksheet on Quadratic Sequence

**Quadratic Sequence**

- Quadratic sequences represent a captivating area of mathematical study, offering a window into the enchanting world of patterns, equations, and numerical relationships.
These sequences, defined by their quadratic nature, are a cornerstone of algebra, serving as a crucial tool for modelling real-world phenomena and solving mathematical puzzles.

In this article, we will discuss:

**What is a Quadratic Sequence?****The formula for Quadratic Sequence?****Finding the nth term of a Quadratic Sequence?**

Here is one more link to practice a few extra questions: Maths Genie Quadratic Sequence Questions

- Quadratic sequences are mathematical sequences where the difference between the terms is not constant.
- In contrast to linear sequences, where the difference is constant, quadratic sequences have constant differences between the differences of their terms.
- This means that the sequence of first differences is a linear sequence.
- Let’s take an example to understand this better:

**Sequence: 1, 4, 9, 16, 25, …**

**1st Differences: 3, 5, 7, 9, …**

**2nd Differences: 2, 2, 2, …**

- As we can see, the differences between the terms are not constant, but the differences between the first differences are constant. Therefore, we can conclude that this is a quadratic sequence.

Question 1: Find the next three terms in the quadratic sequence:** 2, 6, 12, 20, ...**

**Step #1:**To find the next terms, observe the differences between consecutive terms:

**6 - 2 = 4, **

**12 - 6 = 6, **

**20 - 12 = 8.**

**Step #2:**The differences are increasing by 2 each time, so the next differences would be 10, 12, 14.

**Step #3:**Adding these differences to the last term of the sequence:

**20 + 10 = 30, **

**30 + 12 = 42, **

**42 + 14 = 56.**

**Step #4:**Therefore, the next three terms are 30, 42, 56.

Question 2: Find the next three terms in the quadratic sequence:** 4, 13, 25, ...**

**Solution:**

**Step #1:**To find the next terms, observe the differences between consecutive terms:

**13 - 4 = 9, **

**25 - 13 = 12.**

**Step #2:**The differences between consecutive terms are not constant, suggesting a non-linear pattern.**Step #3:**However, let's examine the differences between the differences:

**12 - 9 = 3.**

**Step #4:**The difference between the differences is constant at 3.**Step #5:**Adding this difference to the last difference, we get:

**12 + 3 = 15.**

**Step #6:**Now, adding this difference to the last term of the sequence, we can find the next terms:

**25 + 15 = 40, **

**40 + 15 = 55, **

**55 + 15 = 70.**

**Step #7:**Therefore, the next three terms in the sequence are 40, 55, 70.

- One can recognize and continue a quadratic sequence, and find the formula for the nth term of a quadratic sequence in terms of n.
The formula for the nth term of a quadratic sequence is in the form of

**a _{n} = an^{2} + bn + c.**

We can use the following process to find a, b, and c.

Question 1: Find the missing term in the quadratic sequence: **9, 16, ?, 36, 49.**

**Step #1:**To find the missing term, we observe the differences between consecutive terms:

**16 - 9 = 7, **

**36 - 16 = 20, **

**49 - 36 = 13.**

**Step #2:**We can see that the differences are not constant, so this is not a linear sequence.**Step #3:**However, if we consider the differences between the differences:

**20 - 7 = 13.**

**Step #4:**The difference between the differences is constant at 13.**Step #5:**Adding this difference to the last difference, we get:

**13 + 20 = 33.**

**Step #6:**Now, adding this difference to the last term of the sequence, we can find the missing term:

**36 + 33 = 69.**

**Step #7:**Therefore, the missing term in the sequence is 69.

Question 2: Find the sum of the first 5 terms in the quadratic sequence:** 2, 5, 10, 17, ...**

**Solution:**

**Step #1:**To find the sum of the first 5 terms, we can use the formula for the sum of an arithmetic series.**Step #2:**The nth term of this quadratic sequence can be written as n^{2}+ 1.**Step #3:**Substituting n = 1, 2, 3, 4, 5 into the formula, we get:

**1 ^{2} + 1 = 2, **

**2 ^{2} + 1 = 5, **

**3 ^{2} + 1 = 10, **

**4 ^{2} + 1 = 17, **

**5 ^{2} + 1 = 26**

**Step #4:**The sum of these terms can be found by adding them up:

**2 + 5 + 10 + 17 + 26 = 60**

**Step #5:**Therefore, the sum of the first 5 terms in the sequence is 60.

**Step #1:**** Find the sequence of first and second differences**

- Note: Make sure that the first differences are not constant and the second differences are constant to ensure that it is a quadratic sequence.

Let’s consider the following example:

Sequence: 2, 4, 8, 14, 22, …

1st Differences: 2, 4, 6, 8, …

2nd Differences: 2, 2, 2, …

As the second differences are constant, we can conclude that this is a quadratic sequence

**Step #2: ****Determine a, the second difference, by dividing the second difference by 2. **

- In the above example, a = 2/2 = 1.

**Step #3: Write out the first three or four terms of an ^{2} with the first three or four terms of the **

**an ^{2} = 1, 4, 9, 16, …**

**Step #4: Work out the difference between each term of an ^{2} and the corresponding term of **

**Step #5: Work out the linear nth term of these differences. **

This is bn + c.

In the above example, we get:

**bn + c = n + 2.**

**Step #6: Add this linear nth term to an ^{2} to obtain the nth term of the quadratic sequence.**

In the above example, we have the nth term of the quadratic sequence as

**an ^{2} + bn + c = n^{2} – n + 2**

- Before finding the nth term of a quadratic sequences, compare it to the sequence of square numbers (1, 4, 9, 16, 25, …) and look for a formula.
- For example, in the sequence 4, 7, 12, 19, 28, …, each term is 3 more than the corresponding square number.
- Comparing sequences to known patterns like square numbers can help identify formulas and simplify the process of finding the nth term.

Question 1: Find the nth term of the quadratic sequence**: 6, 15, 28, 45, ...**

**Step #1:**To find the nth term, observe the differences between consecutive terms:

**15 - 6 = 9, **

**28 - 15 = 13, **

**45 - 28 = 17.**

**Step #2:**The differences are increasing by 4 each time.**Step #3:**Therefore, the next differences would be 21, 25, 29.**Step #4:**Now, let's find the differences between the differences:

**13 - 9 = 4, **

**17 - 13 = 4, ...**

**Step #5:**The differences between the differences are constant at 4.**Step #6:**Therefore, the general formula for this quadratic sequence is given by nth term

**= 2n ^{2} + 4n.**

Question 2: Find the missing term in the quadratic sequence:** 2, 8, ?, 32, 50.**

**Solution:**

**Step #1:**To find the missing term, observe the differences between consecutive terms:

**8 - 2 = 6, **

**32 - 8 = 24, **

**50 - 32 = 18.**

**Step #2:**The differences are not constant, indicating a non-linear sequence.**Step #3:**However, consider the differences between the differences:

**24 - 6 = 18.**

**Step #4:**The difference between the differences is constant at 18.**Step #5:**Adding this difference to the last difference, we get:

**18 + 24 = 42.**

**Step #6:**Now, adding this difference to the last term of the sequence, we find the missing term:

**32 + 42 = 74.**

**Step #7:**Therefore, the missing term in the sequence is 74.

**Question 1:** Find the sum of the first 6 terms in the quadratic sequence:** 3, 9, 19, 33, ...**

**Question 2:** Find the nth term of the quadratic sequence:** 1, 6, 15, 28, ...**

**Question 3:** Find the missing term in the quadratic sequence: **4, 13, ?, 37, 52.**

**Question 4:** Find the sum of the first 8 terms in the quadratic sequence:** 2, 7, 16, 29, ...**