Quadratic Sequences

Every quadratic sequence has a position-to-term rule (the nth term) in the form $an^2 + bn + c$, where $a, b,$ and $c$ are constants and $a \neq 0$.

• The constant second difference is always equal to $2a$.
• To find the value of $a$, you simply halve the second difference ($a = \text{second difference} \div 2$).
• The sequence formed by the first differences is always a linear sequence (an arithmetic sequence).

Deriving the nth Term Formula

To find the full $n^{th}$ term formula, you must calculate the values of $a, b,$ and $c$. The most reliable approach is the subtraction method, which breaks the sequence into a quadratic part ($an^2$) and a linear part ($bn + c$).

Subtracting the $an^2$ terms from the original sequence leaves a linear sequence. You can then find the formula for this linear remainder and combine it with your $an^2$ term.

Worked Example: Finding the nth Term

Find an expression, in terms of $n$, for the $n^{th}$ term of the sequence: $1, 10, 23, 40, 61, \dots$

Step 1: Identify the first and second differences to calculate $a$.

• First differences: $10-1=9$, $23-10=13$, $40-23=17$, $61-40=21$
• Second differences: $13-9=4$, $17-13=4$, $21-17=4$
• The constant second difference is $+4$, so $2a = 4 \Rightarrow a = 2$.

Step 2: Generate the $an^2$ sequence.

• Since $a = 2$, calculate $2n^2$ for the first five terms ($n = 1, 2, 3, 4, 5$).
• $2(1)^2=2$, $2(2)^2=8$, $2(3)^2=18$, $2(4)^2=32$, $2(5)^2=50$.

Step 3: Subtract the $an^2$ terms from the original sequence.

• Original sequence: $1, 10, 23, 40, 61$
• Subtract $2n^2$: $(1-2), (10-8), (23-18), (40-32), (61-50)$
• Linear remainder sequence: $-1, 2, 5, 8, 11$

Step 4: Find the $n^{th}$ term of the linear remainder sequence.

• The sequence $-1, 2, 5, 8, 11$ has a common difference of $3$, so $b = 3$.
• The "zeroth" term (the value backwards from the first term) is $-1 - 3 = -4$, so $c = -4$.
• Linear part: $3n - 4$

Step 5: Combine the quadratic and linear parts.

• $n^{th} \text{ term} = 2n^2 + 3n - 4$

Calculating and Verifying Terms

Once you have the $n^{th}$ term formula, you can calculate any term in the sequence by substituting the position number $n$. Remember that $n$ must always be a positive integer.

To verify if a specific number $k$ belongs in the sequence, set the formula equal to $k$ and solve for $n$. If solving the equation results in $n$ being a fraction, decimal, or negative number, then $k$ is not a valid position and the number is not a term in the sequence.

Worked Example: Term Verification

Is 136 a term in the sequence with the $n^{th}$ term $n^2 + 3n + 2$?

Step 1: Set the formula equal to the number.

• $n^2 + 3n + 2 = 136$

Step 2: Rearrange to form a quadratic equation equal to zero.

• $n^2 + 3n - 134 = 0$

Step 3: Solve for $n$ using the quadratic formula.

$$n = \frac{-3 \pm \sqrt{3^2 - 4(1)(-134)}}{2} = \frac{-3 \pm \sqrt{545}}{2}$$

Step 4: State your conclusion based on whether $n$ is an integer.

• Since $\sqrt{545}$ is not a perfect square, $n$ will be a decimal.
• Because $n$ is not a positive integer, 136 is not a term in the sequence.