A quadratic sequence involves a difference which contains a squared term.

Successive values in the sequence can be determined to establish a first difference.

The first difference provides a new sequence. The difference in values in the first sequence generates a second difference. This second difference will be a constant value.

## Example 1

What is the value of the next number in this quadratic sequence?

9, 18, 31, 48, 69, ...

Work out the first difference, then work out the second difference.

The second difference is 4: Add 4 to the last 1st difference: 21 + 4 = 25.

Add 25 to the last value (69) to obtain 94.

 Term 1 2 3 4 5 6 Value 9 18 31 48 69 94 1st Difference 9 13 17 21 25 2nd Difference 4 4 4 4

## Example 2

The sequence shown below has been generated from a quadratic progression. What is the missing number?

-7, -2, ..., 26, 49, 78

Term 1 2 3 4 5 6
Value -7 -2 9 26 49 78
1st Difference 5 11 17 23 29
2nd Difference 6 6 6 6

Because it is a quadratic sequence, the 2nd term differences must be constant. The 2nd difference can be worked out from the last three terms as 6.

Using this, complete all the first differences, then use the first difference to obtain the value of the third term, which is 9.