MathematicsExam code: 0580
Sequences
Algebra and Sequences
Comparing with the squares and the cubes
- A quadratic sequence has n² in its nth term, and its first differences change while the second differences stay equal
- A cubic sequence has n³ in its nth term, and it is the third differences that settle down to a constant
- The fastest route into a straightforward one is to write the square or cube numbers underneath and compare term by term
- A constant gap means adding or subtracting that number, and a constant ratio means multiplying
Exam tip
Write the differences even when you spot the squares
Write the rows of differences down, even when you compare with the square or cube numbers as in the example below. In every scheme for this type, a constant second difference (the third for a cubic) earned one of the two marks on its own; two schemes wanted at least two of its values shown. Any answer of the right type, quadratic or cubic, earned that mark too, so never leave the answer blank.
Worked example
Comparing with a familiar sequence
Find the nth term of 4, 7, 12, 19, 28 and of 3, 10, 29, 66, 127.
Solution:
- Write the square numbers under the first sequence: 1, 4, 9, 16, 25
- Every term is 3 more than the square below it, so the nth term is n² + 3
- Check with n = 5: 25 + 3 = 28, which matches
- Write the cube numbers under the second sequence: 1, 8, 27, 64, 125
- Every term is 2 more than the cube below it, so the nth term is n³ + 2
- Check with n = 4: 64 + 2 = 66, which matches
- A sequence can also be the squares or cubes starting from a later point, in which case the bracket shifts
- 9, 16, 25, 36 are the square numbers from 3² onwards, so the nth term is (n + 2)²