Quadratic sequences

To find the $n$th term of a quadratic sequence, we aim for a formula in the form:

$$a_n=a{n}^2+bn+c$$

Here’s a step-by-step method using an example:

Step 1: Start with the Sequence

Consider the sequence:

$$10,24,44,70,102$$

Step 2: Calculate the Differences

Find the first differences (subtract each term from the next):

$$24-10=14,44-24=20,70-44=26,102-70=32$$

The first differences are: $14,20,26,32$.

Now, find the second differences (subtract each first difference from the next):

$$20-14=6,26-20=6,32-26=6$$

The second differences are constant ($6$), confirming this is a quadratic sequence.

Step 3: Find the Coefficient of $n^2$

Halve the second difference to find the coefficient of $n^2$:

$$a=\frac{6}{2}=3$$

So the $n^2$-term is $3n^2$.

Step 4: Subtract the $n^2$-Term

Subtract $3n^2$ from each term of the sequence to create a new sequence. For $10,24,44,70,102$:

$$10-3(1^2)=7,24-3(2^2)=12,44-3(3^2)=17,70-3(4^2)=22,102-3(5^2)=27$$

The new sequence is $7,12,17,22,27$.

Step 5: Find the Linear $n$th Term

The new sequence $7,12,17,22,27$ is an arithmetic sequence with a common difference of $5$ and a 0th term of $2$. Its formula is:

$$5n+2$$

Step 6: Combine Terms

The quadratic $n$th term is the sum of $3n^2$ and $5n+2$:

$$a_n=3n^2+5n+2$$

Using the Formula

You can use this formula to find any term in the sequence. For example, to find the 4th term ($a_4$):

$$a_4=3(4^2)+5(4)+2=3(16)+20+2=48+20+2=70$$

The 4th term is 70.

Summary

• Find the first and second differences to confirm the sequence is quadratic.
• Halve the second difference to find the coefficient of $n^2$.
• Subtract $3n^2$ (or whatever $a{n}^2$ is) from the sequence to get a linear sequence.
• Find the $n$th term of the linear sequence.
• Add the quadratic and linear terms to get the full formula.