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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

Halve the second difference to get the coefficient of n². Subtract that multiple of the square numbers from the original sequence, and what is left is a linear sequence you already know how to do. "nth term" is asked in 20 of the 43 papers, so this route is worth being quick at.

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