Albert Teen

YOU ARE LEARNING:

Quadratic sequences don't have the same difference each time, but there is a pattern in the differences between consecutive numbers, which we can use to find the nth term.

Quadratic sequences are a type of sequence where there is a common difference between the differences between terms. This is sometimes called a second difference.

For example, let's consider the following sequence:

$2, 6, 12, 20, ....$

1

What is the difference between the first two terms?

2

How about the difference between the second and third terms?

3

And between the third and fourth terms?

4

There is a common difference of $2$ between these differences. So the difference between each term increases by 2 for each new term.

5

This is called the second difference

The second difference of 2 indicates that this is a quadratic sequence.

What is the second difference of this sequence?

$3, 8, 15, 24...$

Let's continue with our sequence: $2, 6, 12, 20$

1

We know that the second difference is $2$

Therefore, each time there is a new term, the difference between it and the previous term increases by 2.

2

So what would the next term be?

The difference between the last two terms is $8$. Since the second difference is $2$, this means that the next term should be $8+2=10$ larger than the last term in the sequence.

3

What is the next term in the sequence?

4

Our sequence is now $2,6, 12, 20, 30$

We can keep adding terms to the end of the sequence, using the second difference of $2$. The next term in the sequence would be $12$ larger than the last, since the difference between $20$ and $30$ is $10$.

5

What is the next term in $2, 6, 12, 20, 30$?

6

The next term is $42$

Nice! đź‘Ť

What would be the next term in this sequence?

$3, 8, 15, 24$

1

The second difference is constant each time