Quadratic And Cubic Sequences

The following figure shows how to derive the formula for the nth term of a quadratic sequence. Scroll down the page for examples and solutions on how to use the formula.

The following figure shows how to derive the formula for the nth term of a cubic sequence. Scroll down the page for examples and solutions on how to use the formula.

• Worksheets to find the Sequence & nth term

  Find the sequence & n^th term
  Find the sequence & n^th term
  Find the n^th term of a sequence with fractions

Finding the nth Term of a Quadratic Sequence
A quadratic sequence has a general form of \(an^2 + bn + c\). The key characteristic is that the second differences are constant

1. Find the differences:
   Write down the sequence.
   Calculate the first differences (the difference between each consecutive term).
   Calculate the second differences (the difference between each consecutive first difference). If the second differences are constant, it’s a quadratic sequence.
   

2. Find the coefficient \(a\):
   The second difference is always equal to \(2a\).
   \(a=\frac{\text{second difference}}{2}\)

3. Let \(3a + b\) = (first value of the first difference) and solve for \(b\).

4. Let \(a + b + c\) = (first term in the sequence) and solve for \(c\).

Finding the nth Term of a Cubic Sequence
A cubic sequence has a general form of \(an^3 + bn^2 + cn + d\). The key characteristic is that the third differences are constant

1. Find the differences:
   Write down the sequence.
   Calculate the first, second, and then the third differences. If the third differences are constant, it’s a cubic sequence.
   

2. Find the coefficient \(a\):
   The third difference is always equal to \(6a\).
   \(a=\frac{\text{third difference}}{6}\)

3. Let \(12a + 2b\) = (first value of the second difference) and solve for \(b\).

4. Let \(7a + 3b + c\) = (first value of the first difference) and solve for \(c\).

5. Let \(a + b + c + d\) = (first term in the sequence) and solve for \(d\).

Videos

How to find the nth term of a quadratic sequence?

When trying to find the nth term of a quadratic sequence, it will be of the form
an² + bn + c
where a, b, c always satisfy the following equations
2a = 2nd difference (always constant)
3a + b = 2nd term - 1st term
a + b + c = 1st term

Example:

1. Find the nth term, T_n of this sequence
   3, 10, 21, 36, 55, …
2. Find the nth term, T_n of this sequence
   0, 7, 20, 39, 64, …

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Nth term of a Quadratic Sequence

GCSE Maths revision Exam paper practice

Example:
(a) The nth term of a sequence is n² - 2n.
Work out the first three terms of the sequence.

(b) Work out the nth term of the sequence 5, 6, 8, 11, 15, …

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Linear and Quadratic Nth Terms of Sequences

GCSE Maths revision Exam paper practice and help
AQA Linked Paired Pilot Methods 2 Higher Jan 2012 Question 11

Example:
(a)Work out the nth term of the sequence
6, 11, 16, 21, 21, …

(b) Work out the nth term of the sequence.
9, 15, 23, 33, 45, …

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Nth Term of a Quadratic Sequence

Find the nth (general) term of a quadratic sequence by using a method of differences.

Example:
What is the nth term of this quadratic sequence?
2, 17, 40, 71, 110, 157, …

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Nth Term of a Cubic Sequence

How to derive the formula for the nth term of a cubic sequence?

Example:
Find nth term of the following cubic sequences
-6, -4, 10, 42, 98, 184, …
17, 59, 143, 287, 509, 827, …

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Nth Term of a Cubic Sequence

Find the nth (general) term of a cubic sequence by using a method of differences.

Example:
What is the nth term of this cubic sequence?
0, -12, -52, -132, -264, -460, …

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