Arithmetic Sequences Revision Notes for Leaving Cert Mathematics

Arithmetic Sequences

What is an arithmetic sequence?

An arithmetic sequence is a special type of number pattern where each term after the first is found by adding the same fixed number to the previous term. This creates a predictable pattern that makes it easy to find any term in the sequence.

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Key characteristics of arithmetic sequences:

Each term increases or decreases by the same amount
The pattern is consistent throughout the entire sequence
You can jump to any position in the sequence using a formula

Essential components

Every arithmetic sequence has two crucial parts that you must identify to fully understand and work with the sequence.

First term (a)

The first term is simply the starting number of the sequence. We use the letter a to represent this value. It tells us where our sequence begins.

Common difference (d)

The common difference is the fixed number that gets added to each term to create the next term. We use the letter d to represent this value.

How to find the common difference: $d = \text{any term} - \text{previous term}$

This means you subtract any term from the term that comes after it.

Looking at these examples helps us understand the pattern:

When d is positive (+3, +4), the sequence increases
When d is negative (-3), the sequence decreases
The first term and common difference completely determine the entire sequence

Finding any term in the sequence

The real power of arithmetic sequences comes from being able to find any term without listing all the terms before it.

The nth term formula: $T_n = a + (n-1)d$

Where:

$T_n$ = the term you want to find
$a$ = the first term
$n$ = the position number of the term you want
$d$ = the common difference

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Why $(n-1)$ ?

Remember that to get to the nth term, you only add the common difference $(n-1)$ times. For example, to get the 3rd term, you add the common difference twice: once to get the 2nd term, then once more to get the 3rd term.

Worked example 1

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Worked Example: Finding Components and Terms

Problem: Find the key components and specific terms for the sequence: 3, 8, 13, ...

Step 1: Find the first term (a) $a = 3$ (this is simply the first number in the sequence)

Step 2: Find the common difference (d) $d = 8 - 3 = 5$ We can verify: $13 - 8 = 5$ ✓

Step 3: Write the general formula $T_n = a + (n-1)d = 3 + (n-1) \times 5 = 3 + 5n - 5 = 5n - 2$

Step 4: Find the 20th term $T_{20} = 5(20) - 2 = 100 - 2 = 98$

Worked example 2

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Worked Example: Complete Sequence Analysis

Problem: For the sequence 7, 10, 13, 16, ...

(i) Find the nth term formula
(ii) Which term equals 97?
(iii) Show that 168 is not a term in this sequence

Solution:

(i) Finding the formula

First term: $a = 7$
Common difference: $d = 10 - 7 = 3$
Formula: $T_n = 7 + (n-1) \times 3 = 7 + 3n - 3 = 3n + 4$

(ii) Finding which term equals 97 Set $T_n = 97$ : $3n + 4 = 97$ $3n = 97 - 4 = 93$ $n = 31$

Therefore, the 31st term equals 97.

(iii) Testing if 168 is in the sequence Set $T_n = 168$ : $3n + 4 = 168$ $3n = 164$ $n = \frac{164}{3} = 54\frac{2}{3}$

Since n must be a whole number (you can't have a fractional term position), 168 is not a term in this sequence.

Exam tips and common traps

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Essential Exam Strategies:

Always check your common difference: Calculate d using at least two different pairs of consecutive terms to make sure it's consistent
Be careful with negative common differences: When d is negative, the sequence decreases. Don't forget the negative sign in your calculations
Verify your nth term formula: Substitute n = 1, 2, 3 back into your formula to check you get the original sequence terms
Check if a number belongs to the sequence: If you get a non-whole number for n, that number is not in the sequence
Read the question carefully: Make sure you're finding what's actually asked - sometimes it's the term value, sometimes it's the position number

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Key Points to Remember:

Definition: An arithmetic sequence adds the same number (common difference) to get each new term
Key formula: $T_n = a + (n-1)d$ where a is the first term and d is the common difference
Finding d: Subtract any term from the next term: $d = T_2 - T_1 = T_3 - T_2$ , etc.
Verification method: For any number to be in the sequence, solving $T_n = \text{that number}$ must give a positive whole number for n
Pattern recognition: Look for the constant difference between consecutive terms - this is what makes it arithmetic

Arithmetic Sequences (Leaving Cert Mathematics): Revision Notes