# Quadratic Sequences: The Nth Term of a Quadratic Number Sequence

Here, we will be finding the nth term of a quadratic number sequence. A quadratic number sequence has nth term = an² + bn + c

**Example 1 **

Write down the nth term of this quadratic number sequence.

-3, 8, 23, 42, 65...

**Step 1:** Confirm the sequence is quadratic. This is done by finding the second difference.

**Sequence **= -3, 8, 23, 42, 65

**1 ^{st} difference** = 11,15,19,23

**2 ^{nd} difference** = 4,4,4,4

**Step 2:** If you divide the second difference by 2, you will get the value of a.

4 ÷ 2 = 2

So the first term of the nth term is 2n²

**Step 3:** Next, substitute the number 1 to 5 into 2n².

**n **= 1,2,3,4,5

**2n²** = 2,8,18,32,50

**Step 4:** Now, take these values (2n²) from the numbers in the original number sequence and work out the nth term of these numbers that form a linear sequence.

**n **= 1,2,3,4,5

**2n²** = 2,8,18,32,50

**Differences ** = -5,0,5,10,15

Now the nth term of these differences (-5,0,5,10,15) is 5n -10.

So b = 5 and c = -10.

**Step 5**: Write down your final answer in the form an² + bn + c.

2n² + 5n -10

**Example 2**

Write down the nth term of this quadratic number sequence.

9, 28, 57, 96, 145...

**Step 1:** Confirm if the sequence is quadratic. This is done by finding the second difference.

**Sequence** = 9, 28, 57, 96, 145...

**1 ^{st} differences** = 19,29,39,49

**2 ^{nd} differences** = 10,10,10

**Step 2:** If you divide the second difference by 2, you will get the value of a.

10 ÷ 2 = 5

So the first term of the nth term is 5n²

**Step 3:** Next, substitute the number 1 to 5 into 5n².

**n **= 1,2,3,4,5

**5n²** = 5,20,45,80,125

**Step 4:** Now, take these values (5n²) from the numbers in the original number sequence and work out the nth term of these numbers that form a linear sequence.

**n = **1,2,3,4,5

**5n² = **5,20,45,80,125

**Differences = **4,8,12,16,20

Now the nth term of these differences (4,8,12,16,20) is 4n. So b = 4 and c = 0.

**Step 5:** Write down your final answer in the form an² + bn + c.

5n² + 4n

## Questions & Answers

Find the nth term of this sequence 2,9,20,35,54?

The first differences are 7, 11, 15, 19.

The second differences are 4.

Half of 4 is 2, so the first term of the sequence is 2n^2.

If you subtract 2n^2 from the sequence you get 0,1,2,3,4 which has the nth term of n - 1

Therefore your final answer will be 2n^2 + n - 1

Helpful 78- Helpful 5
What is the ninth term of this sequence 6,12,20,30,42,56?

The first differences are 6,8,10,12,14. The second difference is 2. Therefore half of 2 is 1 so the first term is n^2. Subtract this from the sequence gives 5,8,11,14,17. The nth term of this sequence is 3n + 2. So the final formula for this sequence is n^2 + 3n + 2.

Helpful 36- Helpful 13

## Comments

this is very helpful , cheers

im still a bit confused how you work out 'c' like how did you get -10 / 0?

where does this difference came from, like the 4,8,12,16,20?

What if the second difference is an odd number?

Very good explanation I will recommend this site to any one who has got problem understanding quadratic sequence

its it the same if you get for example 3n+3n+3n+1 like 3 nth terms do u just go and make a third sequence from your second sequence

what do you do if the number only repeats after 4 differences:

1 1 2 7 21

0 1 5 14 1

1 4 9 2

3 5 3

2 4

What is the equation?

Conceptual undertanding is very important for students to comprehend quadratic sequence. Hence, the logic of determining the terms of a given sequence defined by a quadratic formula should be the starting point.

The procedure helps far better but understanding of the concept per se in changing the sequence to arithmetic number sequence should be greatly emphasised.

dat is just dam cul. luving dis site. helpd me a lot.

quad eqns made easy for us math phobs people.

its epic

brilliant i was having trouble with quadratic sequences and this helped a lot thanks

N is number of term.

Dats cul

In step one there's a mistake in the first difference

it should be 11,15,19,23

thanks for this, had trouble understanding for my maths homework

17