Leaving Cert Sequences & Series (Higher Level): Arithmetic & Geometric Series

Q: What is the difference between a sequence and a series?

A sequence is an ordered list of terms; a series is the sum of the terms of a sequence.

Q: How do I tell if a sequence is arithmetic or geometric?

If each term is found by adding a fixed number it is arithmetic; if each term is found by multiplying by a fixed number it is geometric.

Q: Is sequences and series on Paper 1?

Yes, sequences and series is examined on Paper 1 of Leaving Cert Higher Maths.

Sequences and series cover arithmetic and geometric patterns, finding the nth term (Tn) and the sum of n terms (Sn). It is examined on Paper 1 and links closely to financial maths.

Written by Sean Doyle · Last updated: 2026-06-12

Key facts

An arithmetic sequence has a common difference; a geometric sequence has a common ratio.
Tn gives the value of the nth term; Sn gives the sum of the first n terms.
An infinite geometric series converges only when the common ratio is between -1 and 1.
Sequences and series are examined on Paper 1.

Sequences and Series explained

What is a Sequence?

A sequence is an ordered list of numbers following a specific pattern or rule. Each number in the sequence is called a term.

For example: - $2, 4, 6, 8, 10, \ldots$ (even numbers) - $1, 4, 9, 16, 25, \ldots$ (perfect squares) - $3, 7, 11, 15, 19, \ldots$ (arithmetic sequence)

The position of a term is denoted by $n$ , where $n = 1, 2, 3, \ldots$

Arithmetic Sequences

An arithmetic sequence is a sequence where each term differs from the previous term by a constant amount called the common difference ( $d$ ).

Formula: If the first term is $a$ and the common difference is $d$ , then: $a, \quad a+d, \quad a+2d, \quad a+3d, \quad \ldots$

To find $d$ : Subtract any term from the next term. $d = T_2 - T_1 = T_3 - T_2$

Example: In the sequence $5, 8, 11, 14, \ldots$ - First term: $a = 5$ - Common difference: $d = 8 - 5 = 3$

General Term Formula

The general term (or $n^{\text{th}}$ term) of an arithmetic sequence is: $T_n = a + (n-1)d$

Where: - $T_n$ = the $n^{\text{th}}$ term - $a$ = first term - $n$ = position of the term - $d$ = common difference

This formula allows you to find any term in the sequence without listing all previous terms.

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Key formulas

Name	Formula	Description
Arithmetic Sequence	$T_n = a + (n-1)d$	nth term of arithmetic sequence with first term a and common difference d
Arithmetic Sum	$S_n = \frac{n}{2}[2a + (n-1)d]$	Sum of first n terms of arithmetic sequence
Arithmetic Sum (Alt)	$S_n = \frac{n}{2}(a + l)$	Sum using first term a and last term l
Geometric Sequence	$T_n = ar^{n-1}$	nth term of geometric sequence with first term a and common ratio r
Geometric Sum	$S_n = \frac{a(1-r^n)}{1-r} \quad (r \neq 1)$	Sum of first n terms of geometric sequence
Infinite Geometric Sum	$S_\infty = \frac{a}{1-r} \quad (\|r\| < 1)$	Sum to infinity when \|r\| < 1

Worked examples

Worked example 1

Find the common difference $d$ for the arithmetic sequence: $12, 17, 22, 27, \ldots$

The common difference is found by subtracting consecutive terms: $d = 17 - 12 = 5$ . You can verify: $22 - 17 = 5$ and $27 - 22 = 5$ .

Answer:

Worked example 2

Find the $15^{\text{th}}$ term of the arithmetic sequence with first term $a = 7$ and common difference $d = 3$ .

Using the formula $T_n = a + (n-1)d$ : $T_{15} = 7 + (15-1) \times 3 = 7 + 14 \times 3 = 7 + 42 = 49$

Answer:

Where students lose marks

Confusing the formulas for arithmetic and geometric series.
Using the wrong value of n when counting terms.
Applying the infinite sum formula when the series does not converge.

Frequently asked questions

What is the difference between a sequence and a series?