Triangular, Square, Cube and Fibonacci Sequences

Special Number Sequences

Have you ever noticed how the arrangement of pins in a bowling alley forms a perfect triangle, while the squares on a chessboard create a perfect grid? Mathematics is full of special sequences where numbers follow these specific visual and algebraic patterns. You need to recognise and use three core types of numerical sequences: triangular, square, and cube numbers.

Triangular Number sequences represent the number of dots needed to form an equilateral triangle.
The first five terms are 1, 3, 6, 10, 15.
Each term is generated by adding the next consecutive integer to the previous term (for example, the 4th term is $1+2+3+4 = 10$ ).

Square Number sequences represent the area of a square ( $n \times n$ ).
The first five terms are 1, 4, 9, 16, 25.
The gap between consecutive terms increases by the next odd number ( $+3, +5, +7$ ), giving them a constant second difference of $2$ .
Cube Number sequences represent the volume of a cube ( $n \times n \times n$ ).
The first five terms are 1, 8, 27, 64, 125.
The number $64$ is unique as it is a shared term: it is both a square number ( $8^2$ ) and a cube number ( $4^3$ ). You must also recognise that $10^3 = 1000$ .

Using the nth Term Formulas

The nth term is a position-to-term rule that calculates a term directly from its position ( $n$ ) without needing to know the previous term. For square numbers, the formula is $n^2$ , and for cube numbers, it is $n^3$ .

The triangular nth term formula is slightly more complex:

T_n = \frac{n(n+1)}{2}

Often, sequences are "shifted" versions of these standard rules. For the sequence $4, 7, 12, 19...$ , comparing it to the standard square numbers ( $1, 4, 9, 16...$ ) reveals that every term is $3$ larger, so the formula is $n^2 + 3$ .

Worked Example: Testing a Triangular Number (Higher Tier)

Show that 120 is a triangular number and find its position in the sequence.

Step 1: Set the triangular nth term formula equal to the target number.

$\frac{n(n+1)}{2} = 120$

Step 2: Multiply by 2 to remove the fraction.

$n(n+1) = 240$
$n^2 + n = 240$

Step 3: Rearrange into a quadratic equation equal to zero and factorise.

$n^2 + n - 240 = 0$
$(n+16)(n-15) = 0$

Step 4: Interpret the solutions.

$n = -16$ or $n = 15$
Since a position ( $n$ ) must be a positive integer, $n = 15$ . Therefore, 120 is the 15th triangular number.

Fibonacci-Type Sequences

The spiral of a snail shell and the arrangement of seeds in a sunflower both follow a famous mathematical pattern: the Fibonacci sequence. A Fibonacci Sequence uses a term-to-term rule where each term is the sum of the two preceding terms.

The standard Fibonacci sequence starts: 0, 1, 1, 2, 3, 5, 8, 13, 21...
Every Fibonacci-type sequence requires two seed values to begin.
Any two starting numbers can generate a valid sequence. For example, using $3$ and $7$ gives: $3, 7, 10, 17, 27...$
As the sequence progresses, the ratio of consecutive terms ( $\frac{a_{n+1}}{a_n}$ ) gets closer and closer to the Golden Ratio (approximately $1.618$ ).

Worked Example: Reversing a Fibonacci Sequence

The 3rd and 4th terms of a Fibonacci-type sequence are 7 and 11. Find the 1st and 2nd terms.

Step 1: Use the rule backwards to find the 2nd term (Term 4 - Term 3).

$11 - 7 = 4$

Step 2: Use the rule backwards to find the 1st term (Term 3 - Term 2).

$7 - 4 = 3$

Step 3: State the final sequence.

The sequence begins: 3, 4, 7, 11.

Algebraic Fibonacci Sequences (Higher Tier)

Fibonacci sequences are often presented algebraically in Edexcel exams. If the first two seed values are $a$ and $b$ , the first six terms of the sequence will always follow this pattern:

1st term: $a$
2nd term: $b$
3rd term: $a + b$
4th term: $a + 2b$
5th term: $2a + 3b$
6th term: $3a + 5b$

Worked Example: Finding Unknown Seed Values

The 3rd term of an algebraic Fibonacci sequence is 7 and the 6th term is 29. Find the values of the first term ( $a$ ) and the second term ( $b$ ).

Step 1: Form simultaneous equations using the standard algebraic pattern.

$a + b = 7$
$3a + 5b = 29$

Step 2: Multiply the first equation by 3 to match the $a$ coefficients.

$3(a + b) = 3(7)$
$3a + 3b = 21$

Step 3: Subtract this from the second equation to eliminate $a$ .

$(3a + 5b) - (3a + 3b) = 29 - 21$
$2b = 8$
$b = 4$

Step 4: Substitute $b = 4$ back into the first equation to find $a$ .

$a + 4 = 7$
$a = 3$

Step 5: State the final values.

$a = 3$ , $b = 4$

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Exam Tips

Common Mistake
Students frequently confuse calculating a cube number with multiplying by 3. Remember that $2^3$ is $2 \times 2 \times 2 = 8$ , not $2 \times 3 = 6$ .
Common Mistake
Never try to find a linear formula ( $dn + c$ ) for triangular numbers; their differences are not constant, so they form a quadratic sequence.
3
In 'Show that' Higher Tier questions, method marks (M1) are awarded for setting up equations like $a+b = 7$ . Always write out the algebraic terms even if you cannot finish the simultaneous equations.
4
For Edexcel Papers 2 and 3, use your calculator's $x^2$ and $x^3$ buttons to generate sequence terms quickly, and use the square root and cube root functions to work backwards to find the position $n$ .

Key Terms(8)

Triangular Number: A number that can be represented as an equilateral triangle of dots; it is the sum of the first n natural numbers.
Square Number: The result of multiplying an integer by itself.
Cube Number: The result of multiplying an integer by itself twice.
nth Term: The general rule or algebraic expression that calculates a term in a sequence based entirely on its position.
Position-to-term Rule: A rule that calculates any term in a sequence based on its position without needing to know the previous term.
Fibonacci Sequence: A sequence where each term is generated by adding together the two previous terms.
Term-to-term Rule: A rule that describes how to get from one term in a sequence to the next term.
Seed Values: The first two initial terms required to begin generating a Fibonacci-type sequence.

Previous NoteQuadratic Sequences Next Subtopic: Nth TermNth Term

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Key Terms(8)

Triangular Number: A number that can be represented as an equilateral triangle of dots; it is the sum of the first n natural numbers.
Square Number: The result of multiplying an integer by itself.
Cube Number: The result of multiplying an integer by itself twice.
nth Term: The general rule or algebraic expression that calculates a term in a sequence based entirely on its position.
Position-to-term Rule: A rule that calculates any term in a sequence based on its position without needing to know the previous term.
Fibonacci Sequence: A sequence where each term is generated by adding together the two previous terms.
Term-to-term Rule: A rule that describes how to get from one term in a sequence to the next term.
Seed Values: The first two initial terms required to begin generating a Fibonacci-type sequence.

Triangular, Square, Cube and Fibonacci Sequences

Special Number Sequences

Triangular Number sequences represent the number of dots needed to form an equilateral triangle.
The first five terms are 1, 3, 6, 10, 15.
Each term is generated by adding the next consecutive integer to the previous term (for example, the 4th term is $1+2+3+4 = 10$ ).

Square Number sequences represent the area of a square ( $n \times n$ ).
The first five terms are 1, 4, 9, 16, 25.
The gap between consecutive terms increases by the next odd number ( $+3, +5, +7$ ), giving them a constant second difference of $2$ .
Cube Number sequences represent the volume of a cube ( $n \times n \times n$ ).
The first five terms are 1, 8, 27, 64, 125.
The number $64$ is unique as it is a shared term: it is both a square number ( $8^2$ ) and a cube number ( $4^3$ ). You must also recognise that $10^3 = 1000$ .

Using the nth Term Formulas

The triangular nth term formula is slightly more complex:

T_n = \frac{n(n+1)}{2}

Worked Example: Testing a Triangular Number (Higher Tier)

Show that 120 is a triangular number and find its position in the sequence.

Step 1: Set the triangular nth term formula equal to the target number.

$\frac{n(n+1)}{2} = 120$

Step 2: Multiply by 2 to remove the fraction.

$n(n+1) = 240$
$n^2 + n = 240$

Step 3: Rearrange into a quadratic equation equal to zero and factorise.

$n^2 + n - 240 = 0$
$(n+16)(n-15) = 0$

Step 4: Interpret the solutions.

$n = -16$ or $n = 15$
Since a position ( $n$ ) must be a positive integer, $n = 15$ . Therefore, 120 is the 15th triangular number.

Fibonacci-Type Sequences

The standard Fibonacci sequence starts: 0, 1, 1, 2, 3, 5, 8, 13, 21...
Every Fibonacci-type sequence requires two seed values to begin.
Any two starting numbers can generate a valid sequence. For example, using $3$ and $7$ gives: $3, 7, 10, 17, 27...$
As the sequence progresses, the ratio of consecutive terms ( $\frac{a_{n+1}}{a_n}$ ) gets closer and closer to the Golden Ratio (approximately $1.618$ ).

Worked Example: Reversing a Fibonacci Sequence

The 3rd and 4th terms of a Fibonacci-type sequence are 7 and 11. Find the 1st and 2nd terms.

Step 1: Use the rule backwards to find the 2nd term (Term 4 - Term 3).

$11 - 7 = 4$

Step 2: Use the rule backwards to find the 1st term (Term 3 - Term 2).

$7 - 4 = 3$

Step 3: State the final sequence.

The sequence begins: 3, 4, 7, 11.

Algebraic Fibonacci Sequences (Higher Tier)

Fibonacci sequences are often presented algebraically in Edexcel exams. If the first two seed values are $a$ and $b$ , the first six terms of the sequence will always follow this pattern:

1st term: $a$
2nd term: $b$
3rd term: $a + b$
4th term: $a + 2b$
5th term: $2a + 3b$
6th term: $3a + 5b$

Worked Example: Finding Unknown Seed Values

The 3rd term of an algebraic Fibonacci sequence is 7 and the 6th term is 29. Find the values of the first term ( $a$ ) and the second term ( $b$ ).

Step 1: Form simultaneous equations using the standard algebraic pattern.

$a + b = 7$
$3a + 5b = 29$

Step 2: Multiply the first equation by 3 to match the $a$ coefficients.

$3(a + b) = 3(7)$
$3a + 3b = 21$

Step 3: Subtract this from the second equation to eliminate $a$ .

$(3a + 5b) - (3a + 3b) = 29 - 21$
$2b = 8$
$b = 4$

Step 4: Substitute $b = 4$ back into the first equation to find $a$ .

$a + 4 = 7$
$a = 3$

Step 5: State the final values.

$a = 3$ , $b = 4$

Sign up to continue reading

Get full access to revision notes, key terms, and exam tips for every subtopic.

Exam Tips

Common Mistake
Students frequently confuse calculating a cube number with multiplying by 3. Remember that $2^3$ is $2 \times 2 \times 2 = 8$ , not $2 \times 3 = 6$ .
Common Mistake
Never try to find a linear formula ( $dn + c$ ) for triangular numbers; their differences are not constant, so they form a quadratic sequence.
3
In 'Show that' Higher Tier questions, method marks (M1) are awarded for setting up equations like $a+b = 7$ . Always write out the algebraic terms even if you cannot finish the simultaneous equations.
4
For Edexcel Papers 2 and 3, use your calculator's $x^2$ and $x^3$ buttons to generate sequence terms quickly, and use the square root and cube root functions to work backwards to find the position $n$ .

Key Terms(8)

Triangular Number: A number that can be represented as an equilateral triangle of dots; it is the sum of the first n natural numbers.
Square Number: The result of multiplying an integer by itself.
Cube Number: The result of multiplying an integer by itself twice.
nth Term: The general rule or algebraic expression that calculates a term in a sequence based entirely on its position.
Position-to-term Rule: A rule that calculates any term in a sequence based on its position without needing to know the previous term.
Fibonacci Sequence: A sequence where each term is generated by adding together the two previous terms.
Term-to-term Rule: A rule that describes how to get from one term in a sequence to the next term.
Seed Values: The first two initial terms required to begin generating a Fibonacci-type sequence.

Previous NoteQuadratic Sequences Next Subtopic: Nth TermNth Term

Back to Sequence Types

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(8)

Triangular Number: A number that can be represented as an equilateral triangle of dots; it is the sum of the first n natural numbers.
Square Number: The result of multiplying an integer by itself.
Cube Number: The result of multiplying an integer by itself twice.
nth Term: The general rule or algebraic expression that calculates a term in a sequence based entirely on its position.
Position-to-term Rule: A rule that calculates any term in a sequence based on its position without needing to know the previous term.
Fibonacci Sequence: A sequence where each term is generated by adding together the two previous terms.
Term-to-term Rule: A rule that describes how to get from one term in a sequence to the next term.
Seed Values: The first two initial terms required to begin generating a Fibonacci-type sequence.