Quadratic Sequences. The nth term of a quadratic number sequence.

Here we will 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 you 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 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 you final answer in the form an² + bn + c.

5n² + 4n

For another example on finding the nth term of a quadratic sequence click here.

For more difficult examples click here.