Linear Sequences

A Pattern is a set of numbers, shapes, or objects that follow a specific rule. For example, if you look at the numbers $2, 4, 6, 8, 10,$ ..., you can see that each number is $2$ more than the one before it. This is a simple pattern where the rule is "add $2$ each time."

What is a Sequence?

A Sequence is a type of pattern where we have a list of numbers that follow a certain rule. The numbers in a sequence are called terms.

For example:

The sequence $2, 5, 8, 11$ , ... follows the rule "add $3$ each time." This means the sequence increases, and we say it has a common difference of $3$ .
The sequence $10, 7, 4, 1$ , ... follows the rule "subtract $3$ each time." This means the sequence decreases, and we say it has a common difference of $-3$ . In Maths, understanding sequences is important because they appear in many different problems and topics.

What is a Linear Sequence?

A Linear Sequence is a special type of sequence where the difference between one term and the next is always the same. This difference is called the common difference.

For example:

The sequence $2, 5, 8, 11,$ $2, 5, 8, 11,$ ... has a common difference of $3$ $3$ .
- This means that to go from one term to the next, you always add $3$ .
The sequence $10, 7, 4, 1,$ $10, 7, 4, 1,$ ... has a common difference of $-3.$ $- 3.$
- This means that to go from one term to the next, you subtract $3$ .

Understanding the General Term

The General Term of a sequence is like a formula or a rule that tells you how to find any term in the sequence without having to list all the terms before it. We usually call this general term $T_n$ .

Formula for the General Term of a Linear Sequence:

For a linear sequence, the general term is given by the formula: $T_n = a + (n - 1)d$ where:

$T_n$ is the $n$ th term in the sequence.
$a$ is the first term of the sequence.
$d$ is the common difference (the amount you add or subtract each time).
$n$ is the term number (like $1st, 2nd, 3rd,$ ...).

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Example:

Let's use the formula to find the general term for the sequence $3, 8, 13, 18, ...$

Identify the first term $( a )$ and the common difference $( d )$ :

The first term $a = 3$ .
The common difference $d = 5$ (since each term increases by $5$ ).

Substitute these values into the formula: $T_n = 3 + (n - 1) \times 5$

Simplify the formula: $T_n = 3 + 5n - 5$ $T_n = 5n - 2$

Use the formula to find the $50th$ term:

Now you can find any term in the sequence without writing them all out! For example:
What's the $50th$ term? $T_{50} = 5(50) - 2 = 250 - 2 = 248$

Use the formula to find a specific term:

Let's say we want to find which term in the sequence equals $248$ .
We set the general term equal to $248$ and solve for $n$ : $5n - 2 = 248$
Add $2$ to both sides: $5n = 250$
Divide by $5$ : $n = 50$
This tells us that 248 is the 50th term in the sequence.

Why Setting the General Term Equal to a Value Works

When you set the general term formula equal to a specific value (like $248$ ), you're essentially asking, "For what value of $n$ will the term $T_n$ equal that number?"

This process works because:

The general term formula $T_n = a + (n - 1)d$ gives you the value of the $n$ th term.
By solving the equation, you can "work backwards" to find out which position in the sequence corresponds to that value. This method allows you to identify the term number without needing to list out all the terms, which is especially useful for large sequences.

Why is the General Term Important?

Understanding the general term is super helpful because it lets you:

Find any term in the sequence quickly.
Work backwards to figure out where a certain number is in the sequence.