askvity

How to Find a Cubic nth Term?

Published in Mathematics 4 mins read

To find the nth term of a cubic sequence, you need to determine the cubic expression that generates the sequence. This involves analyzing the differences between consecutive terms and using them to construct the an³ + bn² + cn + d formula.

Steps to Finding the Cubic nth Term

Here's a breakdown of the process:

  1. Identify the Sequence: Recognize that the sequence is cubic. This is generally indicated by the third difference between terms being constant.

  2. Calculate Differences: Find the first, second, and third differences of the sequence.

    • First Difference: The difference between consecutive terms.
    • Second Difference: The difference between consecutive first differences.
    • Third Difference: The difference between consecutive second differences. If the third difference is constant, the sequence is cubic.

  3. Determine the 'a' Value: The coefficient of the n³ term (a) is found by dividing the third difference by 6 (since the third difference of n³ is always 6). So, a = (Third Difference) / 6.

  4. Subtract an³ from the Original Sequence: Create a new sequence by subtracting an³ from each term in your original sequence. This will result in a quadratic sequence.

  5. Find the nth term of the Quadratic Sequence: Use standard methods for finding the nth term of a quadratic sequence. The nth term for a quadratic takes the form bn² + cn + d. This involves finding the first and second differences of the quadratic sequence.

    • The second difference will be constant. Half of this second difference will equal b, the coefficient of the n² term.
    • Now subtract bn² from each term in your quadratic sequence from step 4. The difference between each number in the new sequence is constant; this is the coefficient of the n term.
    • After subtracting cn from the last sequence, each value is constant; this value is d.

  6. Combine the Results: Combine the an³ from step 3 with the quadratic nth term you found in step 5 to form the overall cubic nth term.

    • The general form of the nth term will be: an³ + bn² + cn + d.

Example

Let's say you have the sequence: 2, 11, 34, 77, 146...

  1. Differences:

    Sequence 2 11 34 77 146
    1st Diff 9 23 43 69
    2nd Diff 14 20 26
    3rd Diff 6 6

  2. Calculate 'a': The third difference is 6, so a = 6 / 6 = 1. Therefore, the n³ term is involved.

  3. Subtract an³: Subtract n³ (or 1n³) from each term in the original sequence:

    n 1 2 3 4 5
    n³ 1 8 27 64 125
    Sequence 2 11 34 77 146
    Seq - n³ 1 3 7 13 21

  4. Find nth Term for Quadratic Sequence: The resulting sequence (1, 3, 7, 13, 21) is quadratic.

    Sequence 1 3 7 13 21
    1st Diff 2 4 6 8
    2nd Diff 2 2 2

    b = 2/2 = 1, so the sequence will involve n².

  5. Subtract bn²: Now, subtract 1n² from each term in the quadratic sequence

    n 1 2 3 4 5
    Sequence 1 3 7 13 21
    -n² -1 -4 -9 -16 -25
    Seq - n² 0 -1 -2 -3 -4

  6. The difference between each term is -1 and then we can see that d = 1. This means that the quadratic sequence can be written as: n² - n.

  7. Combine Terms: The nth term of the cubic sequence is then n³ + n² - n.

Key Considerations

  • Accuracy: Double-check your calculations at each step.
  • Practice: Work through various examples to master the process.

By following these steps, you can successfully determine the nth term of a cubic sequence.

Related Articles