How do I Find the Nth Term in a Quadratic Sequence?

Nth Term in a Quadratic Sequence

A quadratic sequence is a sequence where the n^th term rule includes an n² (remember, a term is a word for a number in a sequence). Unlike a linear sequence, the terms in a quadratic sequence do not have a common difference.

A quadratic sequence is a sequence where the n^th term rule includes an n² (remember, a term is a word for a number in a sequence). Unlike a linear sequence, the terms in a quadratic sequence do not have a common difference.

Finding the following term (n^th term) in a quadratic sequence:

Say we wanted to see the next term in the quadratic sequence 9, 15, 23, 33, 45:

The difference between each pair of terms is different, but you may notice the differences themselves form a pattern – they go up by two each time:

The differences between the differences, or the ‘second differences,’ are always the same in a quadratic sequence; this lets us find the next term. So, for example, if we know the second difference is 2, that means the difference between the last term, 45, and the next term must be 12 + 2:

It, in turn, means that the next term must be 45 + 14, or 59:

Finding any term in a quadratic sequence using an n^th term rule:

In any sequence, if you know the n^th term rule, you can find any term in that sequence. Remember, n stands for the position of the term: for the 1st term, n = 1; for the 10th term, n = 10; for the 1765th term, n = 1765. So we substitute n into the n^th term rule for our quadratic sequence. So, for example, say we want to find the 9th term in the sequence with nth term rule 3n² – 5n + 3.

In this case, we are looking for the term 9, so n = 9. We can substitute n = 9 into the n^th term rule:

• n^th term = 3n² – 5n + 3
• 9^th term = 3 × 9² – 5 × 9 + 3

We can work this out step-by-step, remembering to use the order of operations – indices first, then multiplications, then additions and subtractions:

• 9^th term = 3 × 81 – 5 × 9 + 3
• 9^th term = 243 – 45 + 3

Once you’re left with only additions and subtractions, carry them out in the order they are given:

• 9^th term = 198 + 3
• 9^th term = 201

Finding the n^th term rule of a quadratic sequence:

The n^th-term rule of a quadratic sequence can always be written as an² + bn + c. To find this rule, we must find a, b and c. There are a couple of ways to do this, but in either case, we first must find the first and second differences. We’ll start by finding the n^th term rule of the quadratic sequence 6, 15, 28, 45, 66.

The an2 gives the quadratic part of a quadratic sequence in the n^th-term rule. If we remove that, we are left with a linear sequence. We already know n for each term, and a is easy to find: it’s simply the second difference divided by 2. In this case:

• a = 4 ÷ 2 = 2

Now, we can use the value of n and a to find the linear part of this quadratic sequence; this will let us find the values of b and c:

n	1	2	3	4	5
Sequence	6	15	28	45	66
2n²	2	8	18	32	50
Sequence – 2n²	4	7	10	13	16

 

It gives us the linear sequence 4, 7, 10, 13, 16, which has an n^th-term rule of 3n + 1 (a common difference of 3 and the 0^th term of 1). If we combine an² with bn + c, we get the n^th term rule of our quadratic sequence 2n² + 3n + 1.

You might be wondering why, to find the n^th term of a quadratic sequence, we divide the second difference by 2 to see the value of a, when in a linear series, we can use the difference itself. The second method to find the n^th term of a quadratic series should clarify this.

As we said before, the n^th-term rule of any quadratic sequence can be written in the form an² + bn + c. So, let’s substitute the numbers 1 to 5 for n to write out the first five terms of the sequence an² + bn + c.

• If n = 1, an² + bn + c = a + b + c
• If n = 2, an² + bn + c = 4a + 2b + c
• If n = 3, an² + bn + c = 9a + 3b + c
• If n = 4, an² + bn + c = 16a + 4b + c
• If n = 5, an² + bn + c = 25a + 5b + c

It gives us an algebraic quadratic sequence:

a + b + c, 4a + 2b + c, 9a + 3b + c, 16a + 4b + c, 25a + 5b + c

Like any other quadratic sequence, we can find the first and second differences:

It gives us the form of the first five terms, the first differences, and the second differences of any quadratic sequence. Then, we can compare these with a numerical quadratic sequence to find the values of a, b and c. For example, consider the quadratic series we used before:

Focusing on the numbers on the left (as this is where the terms of our algebraic sequence are simplest), we know that 2a and 4 must be equal, we know that 3a + b and 9 must be equal, and we know that a + b + c and 6 must be equal. It gives us three equations to solve:

• 2a = 4
• 3a + b = 9
• a + b + c = 6

Looking at the first equation, we can divide both sides by 2 to get a = 2. It is why we always divide the second difference by 2 to find the value of a. Substituting into the second equation, we get:

• 3a + b = 9
• 3 × 2 + b = 9
• 6 + b = 9
• b = 3

Finally, we can substitute this into the third equation:

• a + b + c = 6
• 2 + 3 + c = 6
• c = 1

It gives us the n^th term rule of our quadratic sequence in the form an² + bn + c:

• 2n² + 3n + 1

Note that the n^th-term rule of a sequence will be the same regardless of the method you use to find it.