Quadratic Sequences Revision Notes for Junior Cert Mathematics

Quadratic Sequences

A Quadratic Sequence is a type of sequence where the difference between the terms isn't always the same, but if you look at the differences between those differences, they will be the same. This may sound a bit confusing at first, so let's break it down.

For example:

Consider the sequence: $4, 7, 12, 19, 28, ...$ $4, 7, 12, 19, 28, ...$
- The first differences (how much the sequence goes up each time) are: $3, 5, 7, 9$ (these aren't the same, so it's not a simple pattern).
- The second differences (the differences between the first differences) are: $2, 2, 2$ (these are the same, which tells us this is a quadratic sequence).

Finding the Formula for a Quadratic Sequence

To find a formula (or general term) that tells you what any term in the sequence will be, we use this formula: $T_n = an^2 + bn + c$ where:

$T_n$ is the term you're trying to find (like the $1st$ term, $2nd$ term, etc.).
$n$ is the position of that term in the sequence.
$a$ , $b$ , and $c$ are numbers we need to figure out by using the sequence.

Steps to Find the General Term:

Find the Second Difference:

Look at the differences between the terms, then look at the differences between those differences. The second difference helps you figure out the value of $a$ . Specifically, $2a$ equals the second difference, so divide the second difference by $2$ to find $a$ .

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Exam Tip: It can be helpful to remember that $a$ is half the second difference.

Create Two Equations:

Use the first two terms of the sequence to create two equations. These equations help you find the values of $b$ and $c$ .

Solve for $b$ and $c$ :

Once you have $a$ , plug it into your equations and solve for $b$ and $c$ .

Write the Formula:

Now that you have $a$ , $b$ , and $c$ , you can write the formula for the sequence.

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Worked Example:

Let's find the formula for the sequence $4, 7, 12, 19, 28, ...$

Find the First and Second Differences: $\begin{array}{c|c|c|c|c} \text{Term number} & 1 & 2 & 3 & 4 & 5 \\ \hline \text{Sequence} & 4 & 7 & 12 & 19 & 28 \\ \hline \text{First difference} & & 3 & 5 & 7 & 9 \\ \hline \text{Second difference} & & & 2 & 2 & 2 \\ \end{array}$

The second difference is $2$ , so $2a = 2$ , which means $a = 1$ .

Create Two Equations Using the First Two Terms:

The general term is $T_n = an^2 + bn + c$ .
Substitute $a = 1$ into the formula: $T_n = n^2 + bn + c$ . Now, use the first two terms in the sequence to make two equations.
For the 1st term $( n = 1 )$ , we know $T_1 = 4$ : $1^2 + b(1) + c = 4 \quad \Rightarrow \quad 1 + b + c = 4 \quad \Rightarrow \quad b + c = 3 \quad \text{(Equation 1)}$
For the $2nd$ term $( n = 2 )$ , we know $T_2 = 7$ : $2^2 + b(2) + c = 7 \quad \Rightarrow \quad 4 + 2b + c = 7 \quad \Rightarrow \quad 2b + c = 3 \quad \text{(Equation 2)}$

Solve the Equations:

Subtract Equation $1$ from Equation $2$ to find $b$ : $(2b + c) - (b + c) = 3 - 3 \quad \Rightarrow \quad :success[b = 0]$
Substitute $b = 0$ back into Equation $1$ to find $c$ : $b + c = 3 \quad \Rightarrow \quad 0 + c = 3 \quad \Rightarrow \quad :success[c = 3]$
Now, we have $a = 1$ , $b = 0$ , and $c = 3$ .

Write the Formula:

Substitute $a$ , $b$ , and $c$ into the general term formula: $:highlight[T_n = n^2 + 3]$
This is the formula for the sequence.

Checking the Formula:

Let's check if the formula works by calculating a few terms:

For the $1st$ term $( n = 1 )$ : $T_1 = 1^2 + 3 = 1 + 3 = 4 \quad :success[\text{(Correct!)}]$
For the $2nd$ term $( n = 2 )$ : $T_2 = 2^2 + 3 = 4 + 3 = 7 \quad :success[\text{(Correct!)}]$
For the $3rd$ term $( n = 3 )$ : $T_3 = 3^2 + 3 = 9 + 3 = 12 \quad :success[\text{(Correct!)}]$ The formula $T_n = n^2 + 3$ correctly gives us the terms in the sequence.

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Exam Tip:

Always double-check your formula by substituting it back into the sequence to make sure it gives the right terms.
Practice with different quadratic sequences to get comfortable with finding $a$ , $b$ , and $c$ . By following these steps, you'll be ready to tackle quadratic sequences on your Junior Cycle Maths exam!

Quadratic Sequences (Junior Cert Mathematics): Revision Notes