Quadratic sequences

Topic summary

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

\[ a_n = an^2 + bn + c \]

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):

\[ 24 – 10 = 14, \quad 44 – 24 = 20, \quad 70 – 44 = 26, \quad 102 – 70 = 32 \]

The first differences are: \( 14, 20, 26, 32 \).

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

\[ 20 – 14 = 6, \quad 26 – 20 = 6, \quad 32 – 26 = 6 \]

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 \( 10, 24, 44, 70, 102 \):

\[ 10 – 3(1^2) = 7, \quad 24 – 3(2^2) = 12, \quad 44 – 3(3^2) = 17, \quad 70 – 3(4^2) = 22, \quad 102 – 3(5^2) = 27 \]

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 \( an^2 \) 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.