TL;DR

A quadratic sequence is one with a constant second difference.
That is the difference of the differences between the terms is always the same.
All quadratic sequences have the form $an^{2}+bn+c$.
The value of $a$ is half the second difference.
To find $b$ and $c$:
- subtract $an^{2}$ from the original sequence;
- find the $n$th term of the remaining linear sequence as $bn+c$

A quadratic sequence is one where the second difference is constant.

The second difference is the sequence formed by finding the difference of the first differences.

Example 1 - Determine that $3,7,13,21,31,…$ is a quadratic sequence.

$$ \begin{array}{rccccccccc} \text{Sequence:} & 3 & & 7 & & 13 & & 21 & & 31 \\ \text{1st Difference:} & & 4 & & 6 & & 8 & & 10 & \\ \text{2nd Difference:} & & & 2 & & 2 & & 2 & & \\ \end{array} $$$3,7,13,21,31,…$ is a quadratic sequence as the second difference is $2$.

Example 2 - Determine that $4,7,14,25,40,…$ is a quadratic sequence.

$$ \begin{array}{rccccccccc} \text{Sequence:} & 4 & & 7 & & 14 & & 25 & & 40 \\ \text{1st Difference:} & & 3 & & 7 & & 11 & & 15 & \\ \text{2nd Difference:} & & & 4 & & 4 & & 4 & & \\ \end{array} $$$4,7,14,25,40,…$ is a quadratic sequence as the second difference is $4$.

Example 3 - Determine that $13,22,37,58,85,…$ is a quadratic sequence.

$$ \begin{array}{rccccccccc} \text{Sequence:} & 13 & & 22 & & 37 & & 58 & & 85 \\ \text{1st Difference:} & & 9 & & 15 & & 21 & & 27 & \\ \text{2nd Difference:} & & & 6 & & 6 & & 6 & & \\ \end{array} $$$13,22,37,58,85,…$ is a quadratic sequence as the second difference is $6$.

Example 4 - Determine that $19,36,51,64,75,…$ is a quadratic sequence.

$$ \begin{array}{rccccccccc} \text{Sequence:} & 19 & & 36 & & 51 & & 64 & & 75 \\ \text{1st Difference:} & & 17 & & 15 & & 13 & & 11 & \\ \text{2nd Difference:} & & & -2 & & -2 & & -2 & & \\ \end{array} $$$19,36,51,64,75,…$ is a quadratic sequence as the second difference is $-2$.

Example 5 - Determine that $3.5,6,9.5,14,19.5,…$ is a quadratic sequence.

$$ \begin{array}{rccccccccc} \text{Sequence:} & 3.5 & & 6 & & 9.5 & & 14 & & 19.5 \\ \text{1st Difference:} & & 2.5 & & 3.5 & & 4.5 & & 5.5 & \\ \text{2nd Difference:} & & & 1 & & 1 & & 1 & & \\ \end{array} $$$3.5,6,9.5,14,19.5,…$ is a quadratic sequence as the second difference is $1$.

Example 6 - Determine that $13,8,5,4,5,…$ is a quadratic sequence.

$$ \begin{array}{rccccccccc} \text{Sequence:} & 13 & & 8 & & 5 & & 4 & & 5 \\ \text{1st Difference:} & & -5 & & -3 & & -1 & & 1 & \\ \text{2nd Difference:} & & & 2 & & 2 & & 2 & & \\ \end{array} $$$13,8,5,4,5,…$ is a quadratic sequence as the second difference is $2$.

All quadratic sequences have a general form of $an^{2}+bn+c$.

The value of $a$ is half the second difference.

Example 1 - Determine the value of $a$ for the quadratic sequence $3,7,13,21,31,…$.

$$ \begin{array}{rccccccccc} \text{Sequence:} & 3 & & 7 & & 13 & & 21 & & 31 \\ \text{1st Difference:} & & 4 & & 6 & & 8 & & 10 & \\ \text{2nd Difference:} & & & 2 & & 2 & & 2 & & \\ \end{array} $$$3,7,13,21,31,…$ is a quadratic sequence as the second difference is $2$.
The value of $a$ is $1$.

Example 2 - Determine the value of $a$ for the quadratic sequence $4,7,14,25,40,…$.

$$ \begin{array}{rccccccccc} \text{Sequence:} & 4 & & 7 & & 14 & & 25 & & 40 \\ \text{1st Difference:} & & 3 & & 7 & & 11 & & 15 & \\ \text{2nd Difference:} & & & 4 & & 4 & & 4 & & \\ \end{array} $$$4,7,14,25,40,…$ is a quadratic sequence as the second difference is $4$.
The value of $a$ is $2$.

Example 3 - Determine the value of $a$ for the quadratic sequence $13,22,37,58,85,…$.

$$ \begin{array}{rccccccccc} \text{Sequence:} & 13 & & 22 & & 37 & & 58 & & 85 \\ \text{1st Difference:} & & 9 & & 15 & & 21 & & 27 & \\ \text{2nd Difference:} & & & 6 & & 6 & & 6 & & \\ \end{array} $$$13,22,37,58,85,…$ is a quadratic sequence as the second difference is $6$.
The value of $a$ is $3$.

Example 4 - Determine the value of $a$ for the quadratic sequence $19,36,51,64,75,…$.

$$ \begin{array}{rccccccccc} \text{Sequence:} & 19 & & 36 & & 51 & & 64 & & 75 \\ \text{1st Difference:} & & 17 & & 15 & & 13 & & 11 & \\ \text{2nd Difference:} & & & -2 & & -2 & & -2 & & \\ \end{array} $$$19,36,51,64,75,…$ is a quadratic sequence as the second difference is $-2$.
The value of $a$ is $-1$.

Example 5 - Determine the value of $a$ for the quadratic sequence $3.5,6,9.5,14,19.5,…$.

$$ \begin{array}{rccccccccc} \text{Sequence:} & 3.5 & & 6 & & 9.5 & & 14 & & 19.5 \\ \text{1st Difference:} & & 2.5 & & 3.5 & & 4.5 & & 5.5 & \\ \text{2nd Difference:} & & & 1 & & 1 & & 1 & & \\ \end{array} $$$3.5,6,9.5,14,19.5,…$ is a quadratic sequence as the second difference is $1$.
The value of $a$ is $\frac{1}{2}$.

Example 6 - Determine the value of $a$ for the quadratic sequence $13,8,5,4,5,…$.

$$ \begin{array}{rccccccccc} \text{Sequence:} & 13 & & 8 & & 5 & & 4 & & 5 \\ \text{1st Difference:} & & -5 & & -3 & & -1 & & 1 & \\ \text{2nd Difference:} & & & 2 & & 2 & & 2 & & \\ \end{array} $$$13,8,5,4,5,…$ is a quadratic sequence as the second difference is $2$.
The value of $a$ is $1$.

To find the values of $b$ and $c$ we start by subtracting $an^{2}$ from the original sequence to get a new sequence.

Example 1 - Determine the difference table for the quadratic sequence $3,7,13,21,31,…$.

$$ \begin{array}{rccccccccc} \text{Sequence:} & 3 & & 7 & & 13 & & 21 & & 31 \\ \text{1st Difference:} & & 4 & & 6 & & 8 & & 10 & \\ \text{2nd Difference:} & & & 2 & & 2 & & 2 & & \\ \end{array} $$$3,7,13,21,31,…$ is a quadratic sequence as the second difference is $2$.
The value of $a$ is $1$.
$$ \begin{array}{r|ccccc} n & 1 & 2 & 3 & 4 & 5 \\ \hline u_n & 3 & 7 & 13 & 21 & 31 \\ an^2 & 1 & 4 & 9 & 16 & 25 \\ \hline u_n - an^2 & 2 & 3 & 4 & 5 & 6 \\ \end{array} $$

Example 2 - Determine the difference table for the quadratic sequence $4,7,14,25,40,…$.

$$ \begin{array}{rccccccccc} \text{Sequence:} & 4 & & 7 & & 14 & & 25 & & 40 \\ \text{1st Difference:} & & 3 & & 7 & & 11 & & 15 & \\ \text{2nd Difference:} & & & 4 & & 4 & & 4 & & \\ \end{array} $$$4,7,14,25,40,…$ is a quadratic sequence as the second difference is $4$.
The value of $a$ is $2$.
$$ \begin{array}{r|ccccc} n & 1 & 2 & 3 & 4 & 5 \\ \hline u_n & 4 & 7 & 14 & 25 & 40 \\ an^2 & 2 & 8 & 18 & 32 & 50 \\ \hline u_n - an^2 & 2 & -1 & -4 & -7 & -10 \\ \end{array} $$

Example 3 - Determine the difference table for the quadratic sequence $13,22,37,58,85,…$.

$$ \begin{array}{rccccccccc} \text{Sequence:} & 13 & & 22 & & 37 & & 58 & & 85 \\ \text{1st Difference:} & & 9 & & 15 & & 21 & & 27 & \\ \text{2nd Difference:} & & & 6 & & 6 & & 6 & & \\ \end{array} $$$13,22,37,58,85,…$ is a quadratic sequence as the second difference is $6$.
The value of $a$ is $3$.
$$ \begin{array}{r|ccccc} n & 1 & 2 & 3 & 4 & 5 \\ \hline u_n & 13 & 22 & 37 & 58 & 85 \\ an^2 & 3 & 12 & 27 & 48 & 75 \\ \hline u_n - an^2 & 10 & 10 & 10 & 10 & 10 \\ \end{array} $$

Example 4 - Determine the difference table for the quadratic sequence $19,36,51,64,75,…$.

$$ \begin{array}{rccccccccc} \text{Sequence:} & 19 & & 36 & & 51 & & 64 & & 75 \\ \text{1st Difference:} & & 17 & & 15 & & 13 & & 11 & \\ \text{2nd Difference:} & & & -2 & & -2 & & -2 & & \\ \end{array} $$$19,36,51,64,75,…$ is a quadratic sequence as the second difference is $-2$.
The value of $a$ is $-1$.
$$ \begin{array}{r|ccccc} n & 1 & 2 & 3 & 4 & 5 \\ \hline u_n & 19 & 36 & 51 & 64 & 75 \\ an^2 & -1 & -4 & -9 & -16 & -25 \\ \hline u_n - an^2 & 20 & 40 & 60 & 80 & 100 \\ \end{array} $$

Example 5 - Determine the difference table for the quadratic sequence $3.5,6,9.5,14,19.5,…$.

$$ \begin{array}{rccccccccc} \text{Sequence:} & 3.5 & & 6 & & 9.5 & & 14 & & 19.5 \\ \text{1st Difference:} & & 2.5 & & 3.5 & & 4.5 & & 5.5 & \\ \text{2nd Difference:} & & & 1 & & 1 & & 1 & & \\ \end{array} $$$3.5,6,9.5,14,19.5,…$ is a quadratic sequence as the second difference is $1$.
The value of $a$ is $\frac{1}{2}$.
$$ \begin{array}{r|ccccc} n & 1 & 2 & 3 & 4 & 5 \\ \hline u_n & 3.5 & 6 & 9.5 & 14 & 19.5 \\ an^2 & 0.5 & 2 & 4.5 & 8 & 12.5 \\ \hline u_n - an^2 & 3 & 4 & 5 & 6 & 7 \\ \end{array} $$

Example 6 - Determine the difference table for the quadratic sequence $13,8,5,4,5,…$.

$$ \begin{array}{rccccccccc} \text{Sequence:} & 13 & & 8 & & 5 & & 4 & & 5 \\ \text{1st Difference:} & & -5 & & -3 & & -1 & & 1 & \\ \text{2nd Difference:} & & & 2 & & 2 & & 2 & & \\ \end{array} $$$13,8,5,4,5,…$ is a quadratic sequence as the second difference is $2$.
The value of $a$ is $1$.
$$ \begin{array}{r|ccccc} n & 1 & 2 & 3 & 4 & 5 \\ \hline u_n & 13 & 8 & 5 & 4 & 5 \\ an^2 & 1 & 4 & 9 & 16 & 25 \\ \hline u_n - an^2 & 12 & 4 & -4 & -12 & -20 \\ \end{array} $$

Finally, we combine the $an^{2}$ with the $n$th term of the linear sequence we have found.

Example 1 - Determine the $n$th term for the quadratic sequence $3,7,13,21,31,…$.

$$ \begin{array}{rccccccccc} \text{Sequence:} & 3 & & 7 & & 13 & & 21 & & 31 \\ \text{1st Difference:} & & 4 & & 6 & & 8 & & 10 & \\ \text{2nd Difference:} & & & 2 & & 2 & & 2 & & \\ \end{array} $$$3,7,13,21,31,…$ is a quadratic sequence as the second difference is $2$.
The value of $a$ is $1$.
$$ \begin{array}{r|ccccc} n & 1 & 2 & 3 & 4 & 5 \\ \hline u_n & 3 & 7 & 13 & 21 & 31 \\ an^2 & 1 & 4 & 9 & 16 & 25 \\ \hline u_n - an^2 & 2 & 3 & 4 & 5 & 6 \\ \end{array} $$The $n$th term of $2,3,4,5,6,…$ is $n+1$.

So the $n$th term is $n^{2}+n+1$.

Example 2 - Determine the $n$th term for the quadratic sequence $4,7,14,25,40,…$.

$$ \begin{array}{rccccccccc} \text{Sequence:} & 4 & & 7 & & 14 & & 25 & & 40 \\ \text{1st Difference:} & & 3 & & 7 & & 11 & & 15 & \\ \text{2nd Difference:} & & & 4 & & 4 & & 4 & & \\ \end{array} $$$4,7,14,25,40,…$ is a quadratic sequence as the second difference is $4$.
The value of $a$ is $2$.
$$ \begin{array}{r|ccccc} n & 1 & 2 & 3 & 4 & 5 \\ \hline u_n & 4 & 7 & 14 & 25 & 40 \\ an^2 & 2 & 8 & 18 & 32 & 50 \\ \hline u_n - an^2 & 2 & -1 & -4 & -7 & -10 \\ \end{array} $$The $n$th term of $2,-1,-4,-7,-10,…$ is $-3n+5$.

So the $n$th term is $2n^{2}-3n+5$.

Example 3 - Determine the $n$th term for the quadratic sequence $13,22,37,58,85,…$.

$$ \begin{array}{rccccccccc} \text{Sequence:} & 13 & & 22 & & 37 & & 58 & & 85 \\ \text{1st Difference:} & & 9 & & 15 & & 21 & & 27 & \\ \text{2nd Difference:} & & & 6 & & 6 & & 6 & & \\ \end{array} $$$13,22,37,58,85,…$ is a quadratic sequence as the second difference is $6$.
The value of $a$ is $3$.
$$ \begin{array}{r|ccccc} n & 1 & 2 & 3 & 4 & 5 \\ \hline u_n & 13 & 22 & 37 & 58 & 85 \\ an^2 & 3 & 12 & 27 & 48 & 75 \\ \hline u_n - an^2 & 10 & 10 & 10 & 10 & 10 \\ \end{array} $$The $n$th term of $10,10,10,10,10,…$ is $10$.

So the $n$th term is $3n^{2}+10$.

Example 4 - Determine the $n$th term for the quadratic sequence $19,36,51,64,75,…$.

$$ \begin{array}{rccccccccc} \text{Sequence:} & 19 & & 36 & & 51 & & 64 & & 75 \\ \text{1st Difference:} & & 17 & & 15 & & 13 & & 11 & \\ \text{2nd Difference:} & & & -2 & & -2 & & -2 & & \\ \end{array} $$$19,36,51,64,75,…$ is a quadratic sequence as the second difference is $-2$.
The value of $a$ is $-1$.
$$ \begin{array}{r|ccccc} n & 1 & 2 & 3 & 4 & 5 \\ \hline u_n & 19 & 36 & 51 & 64 & 75 \\ an^2 & -1 & -4 & -9 & -16 & -25 \\ \hline u_n - an^2 & 20 & 40 & 60 & 80 & 100 \\ \end{array} $$The $n$th term of $20,40,60,80,100,…$ is $20n$.

So the $n$th term is $-n^{2}+20n$.

Example 5 - Determine the $n$th term for the quadratic sequence $3.5,6,9.5,14,19.5,…$.

$$ \begin{array}{rccccccccc} \text{Sequence:} & 3.5 & & 6 & & 9.5 & & 14 & & 19.5 \\ \text{1st Difference:} & & 2.5 & & 3.5 & & 4.5 & & 5.5 & \\ \text{2nd Difference:} & & & 1 & & 1 & & 1 & & \\ \end{array} $$$3.5,6,9.5,14,19.5,…$ is a quadratic sequence as the second difference is $1$.
The value of $a$ is $\frac{1}{2}$.
$$ \begin{array}{r|ccccc} n & 1 & 2 & 3 & 4 & 5 \\ \hline u_n & 3.5 & 6 & 9.5 & 14 & 19.5 \\ an^2 & 0.5 & 2 & 4.5 & 8 & 12.5 \\ \hline u_n - an^2 & 3 & 4 & 5 & 6 & 7 \\ \end{array} $$The $n$th term of $3,4,5,6,7,…$ is $n+2$.

So the $n$th term is $\frac{1}{2}n^{2}+n+2$.

Example 6 - Determine the $n$th term for the quadratic sequence $13,8,5,4,5,…$.

$$ \begin{array}{rccccccccc} \text{Sequence:} & 13 & & 8 & & 5 & & 4 & & 5 \\ \text{1st Difference:} & & -5 & & -3 & & -1 & & 1 & \\ \text{2nd Difference:} & & & 2 & & 2 & & 2 & & \\ \end{array} $$$13,8,5,4,5,…$ is a quadratic sequence as the second difference is $2$.
The value of $a$ is $1$.
$$ \begin{array}{r|ccccc} n & 1 & 2 & 3 & 4 & 5 \\ \hline u_n & 13 & 8 & 5 & 4 & 5 \\ an^2 & 1 & 4 & 9 & 16 & 25 \\ \hline u_n - an^2 & 12 & 4 & -4 & -12 & -20 \\ \end{array} $$The $n$th term of $12,4,-4,12,-20,…$ is $-8n+20$.

So the $n$th term is $n^{2}-8n+20$.

  
      Sequence
      2nd Diff
      \( an^2 \)
      \( u_n - an^2 \)
      \( bn+c \)
      \(n\)th Term
    
      \( 6, 13, 24, 39, \dots \)
      \( +4 \)?

      \( 2n^2 \)?

      \( 4, 5, 6, 7 \)?

      \( n+3 \)?

      \( 2n^2+n+3 \)?

      \( 2, 9, 20, 35, \dots \)
      \( +2 \)?

      \( n^2 \)?

      \( 1, 5, 11, 19 \)?

      \( 4n-3 \)?

      \( n^2+4n-3 \)?

      \( 3, 10, 21, 36, \dots \)
      \( +4 \)?

      \( 2n^2 \)?

      \( 1, 2, 3, 4 \)?

      \( n \)?

      \( 2n^2+n \)?

      \( 1, 4, 9, 16, \dots \)
      \( +2 \)?

      \( n^2 \)?

      \( 0, 0, 0, 0 \)?

      \( 0 \)?

      \( n^2 \)?

      \( 10, 22, 40, 64, \dots \)
      \( +6 \)?

      \( 3n^2 \)?

      \( 7, 10, 13, 16 \)?

      \( 3n+4 \)?

      \( 3n^2+3n+4 \)?

      \( 10, 15, 18, 19, \dots \)
      \( -2 \)?

      \( -n^2 \)?

      \( 11, 19, 27, 35 \)?

      \( 8n+3 \)?

      \( -n^2+8n+3 \)?

      \( 3, 5.5, 9, 13.5, \dots \)
      \( +1 \)?

      \( 0.5n^2 \)?

      \( 2.5, 3.5, 4.5, 5.5 \)?

      \( n+1.5 \)?

      \( 0.5n^2+n+1.5 \)?

      \( 5, 2, -5, -16, \dots \)
      \( -4 \)?

      \( -2n^2 \)?

      \( 7, 10, 13, 16 \)?

      \( 3n+4 \)?

      \( -2n^2+3n+4 \)?

      \( 2, 6.5, 14, 24.5, \dots \)
      \( +3 \)?

      \( 1.5n^2 \)?

      \( 0.5, 0.5, 0.5, 0.5 \)?

      \( 0.5 \)?

      \( 1.5n^2+0.5 \)?

      \( 4, 3.5, 2, -0.5, \dots \)
      \( -1 \)?

      \( -0.5n^2 \)?

      \( 4.5, 5.5, 6.5, 7.5 \)?

      \( n+3.5 \)?

      \( -0.5n^2+n+3.5 \)?

Quadratic $n$th Term

Find the $n$th term for the quadratic sequence below.

Sequence	2nd Diff	\( an^2 \)	\( u_n - an^2 \)	\( bn+c \)	\(n\)th Term
\( 6, 13, 24, 39, \dots \)	\( +4 \) ?	\( 2n^2 \) ?	\( 4, 5, 6, 7 \) ?	\( n+3 \) ?	\( 2n^2+n+3 \) ?
\( 2, 9, 20, 35, \dots \)	\( +2 \) ?	\( n^2 \) ?	\( 1, 5, 11, 19 \) ?	\( 4n-3 \) ?	\( n^2+4n-3 \) ?
\( 3, 10, 21, 36, \dots \)	\( +4 \) ?	\( 2n^2 \) ?	\( 1, 2, 3, 4 \) ?	\( n \) ?	\( 2n^2+n \) ?
\( 1, 4, 9, 16, \dots \)	\( +2 \) ?	\( n^2 \) ?	\( 0, 0, 0, 0 \) ?	\( 0 \) ?	\( n^2 \) ?
\( 10, 22, 40, 64, \dots \)	\( +6 \) ?	\( 3n^2 \) ?	\( 7, 10, 13, 16 \) ?	\( 3n+4 \) ?	\( 3n^2+3n+4 \) ?
\( 10, 15, 18, 19, \dots \)	\( -2 \) ?	\( -n^2 \) ?	\( 11, 19, 27, 35 \) ?	\( 8n+3 \) ?	\( -n^2+8n+3 \) ?
\( 3, 5.5, 9, 13.5, \dots \)	\( +1 \) ?	\( 0.5n^2 \) ?	\( 2.5, 3.5, 4.5, 5.5 \) ?	\( n+1.5 \) ?	\( 0.5n^2+n+1.5 \) ?
\( 5, 2, -5, -16, \dots \)	\( -4 \) ?	\( -2n^2 \) ?	\( 7, 10, 13, 16 \) ?	\( 3n+4 \) ?	\( -2n^2+3n+4 \) ?
\( 2, 6.5, 14, 24.5, \dots \)	\( +3 \) ?	\( 1.5n^2 \) ?	\( 0.5, 0.5, 0.5, 0.5 \) ?	\( 0.5 \) ?	\( 1.5n^2+0.5 \) ?
\( 4, 3.5, 2, -0.5, \dots \)	\( -1 \) ?	\( -0.5n^2 \) ?	\( 4.5, 5.5, 6.5, 7.5 \) ?	\( n+3.5 \) ?	\( -0.5n^2+n+3.5 \) ?

Lesson 5 - Quadratic Sequences

TL;DR

Quadratic \(n\)th Term