Strand 4 — Algebra

Sequences: Linear, Quadratic, and Exponential Patterns

1st Year · 2nd Year · 3rd Year (Junior Cert)

✓By the end of this lesson students will be able to identify linear, quadratic, and exponential sequences.
✓By the end of this lesson students will be able to find the next term(s) in a given sequence.
✓By the end of this lesson students will be able to determine the general term (n-th term) for linear sequences.
✓By the end of this lesson students will be able to determine the general term (n-th term) for simple quadratic sequences.
✓By the end of this lesson students will be able to solve problems involving sequences.

Key concepts

Sequence

A sequence is an ordered list of numbers, called terms, that follow a particular pattern or rule.

Term

Each number in a sequence is called a term. We often denote the first term as T₁, the second term as T₂, and so on. The n-th term is denoted as Tₙ, which is also known as the general term.

Linear Sequence (Arithmetic Sequence)

A linear sequence is a sequence where the difference between consecutive terms is constant. This constant difference is called the common difference (d). Each term is found by adding the common difference to the previous term.

Tₙ = a + (n-1)d, where 'a' is the first term and 'd' is the common difference.

Quadratic Sequence

A quadratic sequence is a sequence where the first differences between consecutive terms are not constant, but the second differences (the differences between the first differences) are constant. The general term of a quadratic sequence is of the form Tₙ = an² + bn + c, where a, b, and c are constants.

Tₙ = an² + bn + c

Exponential Sequence (Geometric Sequence)

An exponential sequence is a sequence where each term is found by multiplying the previous term by a constant value. This constant value is called the common ratio (r).

Tₙ = arⁿ⁻¹, where 'a' is the first term and 'r' is the common ratio.

Key facts to remember

1A sequence is an ordered list of numbers following a rule.
2A linear sequence has a constant first difference (common difference, d). Its general term is Tₙ = a + (n-1)d.
3A quadratic sequence has a constant second difference. Its general term is Tₙ = an² + bn + c.
4An exponential sequence has a constant common ratio (r), found by dividing consecutive terms (Tₙ / Tₙ₋₁).
5To find the general term of a quadratic sequence, use the relationships: 2a = second difference, 3a+b = (T₂-T₁), and a+b+c = T₁.
6Always check the differences first. If constant, it's linear. If not, check second differences. If constant, it's quadratic. If neither, check ratios for exponential.

Worked examples

Example 1

Consider the sequence: 5, 9, 13, 17, ... (a) Find the next two terms. (b) Determine the general term (Tₙ) for this sequence.

IPart (a): Find the difference between consecutive terms:

II9 - 5 = 4

III13 - 9 = 4

IV17 - 13 = 4

VThe common difference (d) is 4. This is a linear sequence.

VITo find the next two terms, add 4 to the last given term:

VII17 + 4 = 21

VIII21 + 4 = 25

9Part (b): Use the formula for the general term of a linear sequence: Tₙ = a + (n-1)d

10The first term (a) = 5.

11The common difference (d) = 4.

12Substitute these values into the formula:

13Tₙ = 5 + (n-1)4

14Tₙ = 5 + 4n - 4

15Tₙ = 4n + 1

Answer

(a) The next two terms are 21 and 25. (b) The general term is Tₙ = 4n + 1.

Always check your general term by substituting n=1, 2, 3 to see if it generates the original sequence.

Example 2

Consider the sequence: 3, 8, 15, 24, ... (a) Find the next two terms. (b) Determine the general term (Tₙ) for this sequence.

IPart (a): Find the first differences between consecutive terms:

II8 - 3 = 5

III15 - 8 = 7

IV24 - 15 = 9

VThe first differences are 5, 7, 9. These are not constant, so it's not a linear sequence.

VIFind the second differences:

VII7 - 5 = 2

VIII9 - 7 = 2

9The second differences are constant (2), so this is a quadratic sequence.

10To find the next first difference, add 2 to the last first difference: 9 + 2 = 11.

11To find the next term, add this new first difference to the last term: 24 + 11 = 35.

12To find the next first difference: 11 + 2 = 13.

13To find the next term: 35 + 13 = 48.

14Part (b): Use the general form for a quadratic sequence: Tₙ = an² + bn + c.

15We know that 2a = the constant second difference. So, 2a = 2 => a = 1.

16We know that 3a + b = the first of the first differences (the difference between T₁ and T₂). So, 3(1) + b = 5 => 3 + b = 5 => b = 2.

17We know that a + b + c = the first term (T₁). So, 1 + 2 + c = 3 => 3 + c = 3 => c = 0.

18Substitute a=1, b=2, c=0 into Tₙ = an² + bn + c:

19Tₙ = (1)n² + (2)n + 0

20Tₙ = n² + 2n

Answer

(a) The next two terms are 35 and 48. (b) The general term is Tₙ = n² + 2n.

The relationships 2a = second difference, 3a+b = first difference (T₂-T₁), and a+b+c = T₁ are crucial for finding a, b, and c in quadratic sequences.

Example 3

Consider the sequence: 2, 6, 18, 54, ... (a) Find the next two terms. (b) Identify the type of sequence.

IPart (a): Check for a common difference:

II6 - 2 = 4

III18 - 6 = 12

IVThe differences are not constant, so it's not a linear sequence.

VCheck for a common ratio (divide consecutive terms):

VI6 / 2 = 3

VII18 / 6 = 3

VIII54 / 18 = 3

9The common ratio (r) is 3. This is an exponential sequence.

10To find the next two terms, multiply the last given term by the common ratio:

1154 × 3 = 162

12162 × 3 = 486

13Part (b): Since there is a common ratio, the sequence is an exponential sequence.

Answer

(a) The next two terms are 162 and 486. (b) This is an exponential sequence.

For exponential sequences, the common ratio is found by dividing any term by its preceding term (Tₙ / Tₙ₋₁).

Common mistakes

✗Confusing the methods for finding the general term of linear and quadratic sequences.
✗Calculation errors when finding differences or ratios between terms.
✗Incorrectly simplifying the general term formula for linear sequences (e.g., Tₙ = a + (n-1)d becoming a + nd - d instead of a - d + nd).
✗Assuming a sequence is linear or quadratic without checking the differences properly.
✗Not showing all steps when deriving the general term for quadratic sequences, leading to errors in 'a', 'b', or 'c'.

Exam tips

★Always show your working for finding differences or ratios. This helps you identify the type of sequence and can earn partial marks.
★When finding the general term, substitute n=1, n=2, etc., into your derived formula to check if it generates the original sequence terms.
★Read the question carefully to determine if you need to find the next terms, the general term, or both.
★If a sequence doesn't immediately look linear or quadratic, try dividing consecutive terms to check for an exponential pattern.

Ready to practise?

Try a problem on this topic

Snap a photo or type a question — get step-by-step working instantly.

Solve a problem →← Maths curriculum