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How to Find the nth Term of a Quadratic Sequence

Published in Quadratic Sequences 3 mins read

Finding the nth term of a quadratic sequence involves a systematic approach using differences. A quadratic sequence is a sequence of numbers where the second difference between consecutive terms is constant. This means the difference between consecutive terms themselves forms an arithmetic sequence.

Steps to Finding the nth Term

Here's a step-by-step guide, incorporating the provided reference information:

  1. Find the First and Second Differences: Calculate the differences between consecutive terms (first difference, d1). Then, calculate the differences between the first differences (second difference, d2). If the second difference is constant, you have a quadratic sequence. (Reference: Third Space Learning, BBC Bitesize, Siyavula)

  2. Halve the Second Difference: Divide the constant second difference by 2. This value (let's call it 'a') will be the coefficient of n² in the nth term formula. (Reference: Third Space Learning)

  3. Subtract an² from the Original Sequence: Subtract an² (where 'a' is the value calculated in step 2 and 'n' represents the term number) from each corresponding term in the original sequence. (Reference: Third Space Learning)

  4. Find the nth Term of the Linear Sequence: The result from step 3 should form a linear sequence (a sequence with a constant difference between consecutive terms). Find the nth term of this linear sequence using the formula bn + c, where 'b' is the common difference and 'c' is the y-intercept (the value when n=0). (Reference: Third Space Learning)

  5. Combine to Find the Quadratic nth Term: The nth term of the original quadratic sequence is the sum of the results from steps 2 and 4: an² + bn + c.

Example

Let's consider the sequence: 2, 5, 10, 17, 26...

  1. Differences:

    • First difference (d1): 3, 5, 7, 9,...
    • Second difference (d2): 2, 2, 2,... (constant, confirming it's quadratic)

  2. Halve the Second Difference: a = 2 / 2 = 1

  3. Subtract an²:

    • 2 - 1(1)² = 1
    • 5 - 1(2)² = 1
    • 10 - 1(3)² = 1
    • 17 - 1(4)² = 1
    • 26 - 1(5)² = 1
      This gives the linear sequence: 1, 1, 1, 1, 1...

  4. Linear nth Term: The nth term of this linear sequence is simply 1 (b=0, c=1).

  5. Quadratic nth Term: Therefore, the nth term of the original quadratic sequence is 1n² + 0n + 1 = n² + 1.

Resources for Further Learning

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