Algebra

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Quadratic Sequences: Finding the nth Term

We can construct expressions to help us find any value within a quadratic sequence.

We can also construct expressions which describe the nth term of a quadratic sequence.

Quadratic sequences are named this way because their nth term will be of the form below, where $a$ , $b$ and $c$ are numbers to be found. Notice that this is the same form as quadratic equations!

$an^2 + bn + c$

Looking at the general form, $a^2+bn+c$ , $a$ is always equal to half of the second difference. Let's start with an example:

$2, 6, 12, 20, 30$

What is the second difference here?

$2, 6, 12, 20, 30$

We can find the nth term directly from a sequence. Let's find the nth term for: $2, 6, 12, 20, 30$

We know the second difference is $2$

We also know that $a$ is half of the second difference, so in this case $a=1$ .

The nth term will be $n^2 + bn + c$ .

To find b and c, we can substitute $n = 2$ and $n = 3$ into $n^2 + bn + c$ . Remember that n relates to the position of the number in the sequence.

The second term is 6, where $n=2$

By substituting $n=2$ into the general equation where $a=1$ , we find that $2^2 + 2b + c=6$ . We can then subtract 4 from both sides to find that $2b+c=2$ .

Substitute $n=3$ into $n^2+bn+c=12$ and simplify to the form $ab+c=d$

Now we have two equations

$(1)\space 2b + c = 2$ and $(2)\space 3b + c = 3$ . We can solve these simultaneously!

Subtract equation (1) from equation (2)

$3b-2b+c-c=3-2$ leaves $b=1$

Substitute $b=1$ into equation (1)

$2\times 1 + c=2$ leaves $c=0$

Great work!

$b=1$ and $c=0$ . Therefore, we know that our nth term is found by $n^2+n$ (from looking at our general equation $an^2+bn+c$ ).

We can use this to find any term in the sequence

By substituting the position of the term into the equation, we can find the value of the term.

What is the 10th term in the sequence? $n^2+n$

The following sequence has an nth term of the form $an^2 + c$ . Find the nth term

$5, 14, 29, 50, 77$

You might be given the general form of the nth term, from which you need to find the exact version.

For example, the following sequence has an nth term of the form $n^2 + c$ . Find the nth term.

$8, 11, 16, 23, 32$

The second difference is 2

The difference between the first two terms is $11-8=3$ , between the second two terms is $16-11=5$ , and between the third couple of terms is $23-16=7$ . The common difference between these is two.

Therefore, the coefficient of $n^2$ is 1

As a result, we can be sure that the sequence has an nth term in the form stated: $n^2+c$

We know that at $n=1$ , $n^2+c=8$

The first term in the sequence is 8, so the nth term equals 8 when $n=1$ . Therefore, we can substitute these values into the equation, leaving us with $1^2+c=8$

Rearrange to make $c$ the subject

We can subtract $1$ from both sides here to isolate $c$ . ${\cancel{1^2}}+c=8-1$ so $c=7$ .

Our nth term is $n^2+7$

Using this, we are able to find any term within the sequence.

The following sequence has an nth term of the form $n^2 + c$ . Find the nth term.

$4, 7, 12, 19, 28$