Quadratic Progressions

Quadratic Progressions

GCSE(F), GCSE(H),

A linear sequence involves the difference between each term, and will be a constant value. A quadratic sequence involves looking at the difference between each term, called the first difference which will be a changing value. Then examine the differences between the numbers from the first difference to obtain a second difference, which will be a constant value.

Examples

1. What is the value of the next number in this sequence?
9, 18, 31, 48, 69, ...

Answer: 94

Term12 3456
Value918 31486994
1st Difference913 172125
2nd Difference4 444

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.

2. The sequence shown below has been generated from a quadratic progression. What is the missing number?
-7, -2, ..., 26, 49, 78

Answer: 9

Term12 3456
Value-7-2 ...264978
1st Difference5... ...2329
2nd Difference... ......6

Because it is a quadratic sequence, the 2nd term differences must be constant. From the last thre terms, work out the second difference 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.