n-th Terms of Quadratic Sequences

n-th Terms of Quadratic Sequences

Understanding Quadratic Sequences

  • A quadratic sequence is a sequence in which the second differences between terms are constant. This signifies that the terms follow a pattern based on a quadratic formula.

  • To find the n-th term of a quadratic sequence, you will need to express the sequence in the form an^2 + bn + c.

Determining the Coefficients of the Quadratic Sequence

  • The coefficient a in the n-th term is usually half of the second difference in the sequence.

  • To find the coefficient b, subtract an^2 from each term in the original sequence consequently forming a linear sequence. The common difference in this new sequence will be b.

  • To find the coefficient c, substitute n = 1 into the expression an^2 + bn + c and set it equal to the first term of the sequence. Subsequently, solve for c.


  • After finding a, b, and c, you have the n-th term rule for the quadratic sequence. It allows you to find any term in the sequence without listing all the preceding terms.

  • Example: If a sequence has a second difference of 4 and starts with 2, 6, 12, the n-th term rule is 2n^2 - n.


  • Always double-check your coefficients, as even small mistakes can lead to incorrect results.
  • Other sequences such as cubic sequences or Fibonacci sequences require different approaches.