The Fibonacci sequence and other special number sequences appear in nature, art, and problem-solving. Unlike arithmetic sequences, they cannot always be described by a simple formula — each term may depend on earlier terms, and recognising the underlying pattern is the key skill at KS3. Knowing these sequences by sight saves real exam time.
What is the Fibonacci sequence?
The Fibonacci sequence begins 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 …
The rule: each term is the sum of the two terms before it.
- Term 3: 1 + 1 = 2
- Term 4: 1 + 2 = 3
- Term 5: 2 + 3 = 5
- Term 6: 3 + 5 = 8
This type of rule — where you need at least two previous terms to find the next — is called a recurrence relation. There is no simple formula for the nth term in the form you use for arithmetic or geometric sequences (though one does exist involving surds, which is A-level content).
Fibonacci numbers appear in the spiral arrangements of seeds in a sunflower, the number of petals on many flowers, and the branching patterns of trees. Recognising the sequence is often more useful than deriving it.
What are triangular numbers?
Triangular numbers are found by adding consecutive whole numbers starting from 1:
| Term (n) | Sum | Triangular number |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 1 + 2 | 3 |
| 3 | 1 + 2 + 3 | 6 |
| 4 | 1 + 2 + 3 + 4 | 10 |
| 5 | 1 + 2 + 3 + 4 + 5 | 15 |
| 6 | 1 + 2 + 3 + 4 + 5 + 6 | 21 |
The nth triangular number formula is: Tₙ = n(n + 1)/2.
For example, T₆ = 6 × 7/2 = 42/2 = 21. ✓
Triangular numbers can be pictured as dots arranged in equilateral triangles. The differences between consecutive triangular numbers are 2, 3, 4, 5, 6 … — the sequence of differences grows by 1 each time, which is the hallmark of a second-difference pattern.
What are square numbers in a sequence?
Square numbers are a sequence in their own right: 1, 4, 9, 16, 25, 36, 49 …
The nth square number is simply n². The first differences are 3, 5, 7, 9 … (odd numbers), and the second differences are all 2. This second-difference pattern of 2 identifies a quadratic sequence.
The link between triangular and square numbers: the sum of the first n odd numbers equals n². For example, 1 + 3 + 5 + 7 = 16 = 4². Another link: two consecutive triangular numbers always add up to a square number: T₃ + T₄ = 6 + 10 = 16 = 4².
What are the powers of 2 and other geometric-style sequences?
The powers of 2 — 1, 2, 4, 8, 16, 32, 64, 128 … — form a geometric sequence with first term 1 and common ratio 2. The nth term is 2ⁿ⁻¹.
Similar special sequences include:
- Powers of 3: 1, 3, 9, 27, 81 … (nth term: 3ⁿ⁻¹)
- Powers of 10: 1, 10, 100, 1000 … (links directly to place value and standard form)
- Powers of 1/2: 1, 1/2, 1/4, 1/8 … (a decreasing geometric sequence with r = 1/2)
These appear in problems about standard form, compound interest, and binary arithmetic (computing).
How do you recognise which type of special sequence you have?
A systematic approach works well. Work through these checks in order:
- Is each term the sum of the two before it? → Fibonacci (or a Fibonacci-type sequence with different starting values).
- Is each term found by adding a constant? → Arithmetic sequence (not "special" in the sense above, but common in papers).
- Is each term found by multiplying by a constant? → Geometric sequence (powers of 2, 3, etc.).
- Are the first differences the positive integers 1, 2, 3, 4 …? → Triangular numbers.
- Are the first differences odd numbers (1, 3, 5, 7…) and second differences all 2? → Square numbers.
- Does the sequence grow faster than any of the above? → Could be cube numbers (1, 8, 27, 64 …) or another power.
In exam questions, the first five terms are usually given; spotting which pattern fits within about 20 seconds is the key skill.
What is a Fibonacci-type sequence with different starting values?
You can start a Fibonacci-type sequence with any two numbers and apply the same "add the previous two terms" rule. For example, starting with 2 and 5: 2, 5, 7, 12, 19, 31, 50 … These are not "the Fibonacci sequence" but they follow the same recurrence rule and can appear in KS3 and GCSE questions under labels like "a sequence where each term is the sum of the two preceding terms".
Frequently asked questions
Why does the Fibonacci sequence appear in nature?
Fibonacci numbers arise in nature because many biological growth processes add previous generations together — for example, the way rabbit populations grow if each pair matures after one month and then produces a new pair each month thereafter. Spiral leaf arrangements and seed packings optimise packing efficiency, and the Fibonacci ratio naturally produces the golden angle, approximately 137.5°, that maximises spacing between leaves. This is genuinely beautiful mathematics rather than a coincidence.
What is the connection between Fibonacci numbers and the golden ratio?
As you go further along the Fibonacci sequence, the ratio of consecutive terms (such as 89/55 = 1.618…, 144/89 = 1.618…) gets ever closer to the golden ratio, φ ≈ 1.618. The golden ratio appears in art, architecture, and the proportions of many natural objects. This link is usually encountered at A-level or in enrichment activities, not as assessed KS3 content, but it is worth knowing.
How do I find the 10th triangular number quickly?
Use the formula: Tₙ = n(n + 1)/2. For n = 10: T₁₀ = 10 × 11 / 2 = 110 / 2 = 55. Notice that 55 is also the 10th Fibonacci number — a remarkable coincidence (or perhaps not) that delights mathematicians.
Can any number be triangular, square, and cubic at the same time?
The number 1 satisfies all three: T₁ = 1, 1² = 1, 1³ = 1. The next number that is both triangular and square is 36 (T₈ = 36 = 6²). Finding numbers that are simultaneously triangular, square, and cubic is a much harder number theory problem — 1 is the only known such number up to very large values, though the general question remains open for research mathematicians.
For pattern recognition and sequences practice with Professor Pi, visit aitutors.me.