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Number Sequences Guide

What do I need to know about Arithmetic Sequences?

Arithmetic sequence: constant difference d between consecutive terms. nth term: aₙ = a + (n-1)d. Where a = first term, d = common difference. Sum of n terms: Sₙ = n/2 × (2a + (n-1)d) = n/2 × (first term + last term). Example: 3, 7, 11, 15... d=4, a=3. nth term: 3 + (n-1)×4 = 4n - 1. 20th term: 4×20 - 1 = 79. Sum to 20: 20/2 × (3+79) = 10×82 = 820. Arithmetic sequences describe: salary increases by a fixed amount each year. Equally spaced data points. Simple interest accumulation.

What should I know about Geometric Sequences?

Geometric sequence: constant ratio r between consecutive terms. nth term: aₙ = ar^(n-1). Sum of n terms: Sₙ = a(1-rⁿ)/(1-r) for r≠1. Infinite sum (|r| < 1): S∞ = a/(1-r). Example: 2, 6, 18, 54... r=3, a=2. nth term: 2×3^(n-1). 10th term: 2×3⁹ = 2×19683 = 39,366. Sum to 10: 2×(1-3¹⁰)/(1-3) = 2×(-59048)/(-2) = 59,048. Geometric sequences describe: compound interest. Exponential population growth. Radioactive decay. Computer algorithm complexity O(2ⁿ).

What do I need to know about Quadratic Sequences?

Quadratic sequence: constant second difference. First differences vary, but the differences of the differences are constant. nth term: aₙ = an² + bn + c. Finding the formula: second difference = 2a → find a. Substitute two known terms to find b and c. Example: 1, 4, 9, 16, 25... First differences: 3, 5, 7, 9... Second differences: 2, 2, 2... (constant). Second difference = 2a = 2 → a=1. So n² fits exactly. General quadratic: when second difference is k, the n² coefficient is k/2.

What's the key thing to understand about Fibonacci and Special Sequences?

Fibonacci: 1, 1, 2, 3, 5, 8, 13, 21, 34... Each term = sum of previous two. Golden ratio: limit of consecutive Fibonacci ratios → φ = (1+√5)/2 ≈ 1.618. Appears in: plant spiral patterns (phyllotaxis), shell spirals, art and architecture proportions. Triangular numbers: 1, 3, 6, 10, 15... nth term: n(n+1)/2. Sum of first n natural numbers. Square numbers: 1, 4, 9, 16... nth term: n². Cube numbers: 1, 8, 27, 64... nth term: n³. Prime numbers: 2, 3, 5, 7, 11, 13... No closed-form nth term formula exists for generating primes directly, which is precisely why finding large prime numbers remains a genuinely hard computational problem.

What do I need to know about Finding the nth Term Formula?

For an arithmetic sequence, the nth term is a + (n−1)d, where a is the first term and d is the common difference. For 5, 8, 11, 14...: a=5, d=3, so nth term = 5+(n−1)×3 = 3n+2. Check: 3(4)+2=14 ✓. For a geometric sequence, the nth term is a×r^(n−1): for 3, 6, 12, 24..., a=3, r=2, nth term = 3×2^(n−1). For a quadratic sequence with second difference 2c, the nth term includes cn² plus a linear adjustment found by subtracting cn² from the original sequence and treating what's left as a new, simpler arithmetic sequence. Spotting which type you have is the first step: constant difference means arithmetic, constant ratio means geometric, and a constant second difference (but non-constant first difference) means quadratic.

What should I know about Triangular and Square Numbers?

Triangular numbers (1, 3, 6, 10, 15, 21...) count dots arranged in a triangle and follow nth term = n(n+1)/2 — the same formula as summing 1 to n. Square numbers (1, 4, 9, 16, 25...) follow nth term = n². A striking identity: any two consecutive triangular numbers sum to a square number — 6+10=16=4², 10+15=25=5². Triangular numbers appear in handshake problems (how many handshakes among n people = T(n−1)) and in combinations (C(n,2) = T(n−1)). Square numbers appear whenever something scales in two dimensions, like tiles needed for an n×n grid. Recognising these named sequences quickly — rather than deriving a formula from scratch each time — speeds up GCSE and verbal-reasoning style pattern questions enormously.

What do I need to know about Second Differences and Identifying Sequence Type?

A fast way to classify a number pattern is to look at its differences. If the differences between consecutive terms are constant, the sequence is arithmetic (nth term is linear in n). If those first differences themselves change by a constant amount — the second differences are constant — the sequence is quadratic, and that constant second difference equals 2a in an nth term formula of the form an² + bn + c. For example, 2, 5, 10, 17, 26 has first differences 3, 5, 7, 9 and second differences all equal to 2, confirming a quadratic rule with a = 1 (since 2a = 2); working through the rest gives nth term = n² + 1. If neither first nor second differences are constant but each term is a fixed ratio of the one before, the sequence is geometric instead.

What should I know about a Sequence Type Reference Table?

A quick way to classify any number pattern and jump straight to its nth-term formula:

What should I know about a Worked Example: A Cubic Sequence?

Some sequences require a third level of differencing before a constant appears. Take the sequence 2, 11, 34, 77, 146, ... First differences: 9, 23, 43, 69. Second differences: 14, 20, 26. Third differences: 6, 6 — constant at last, confirming this is a cubic sequence. A constant third difference of k corresponds to a cubic nth term of the form an³+bn²+cn+d, where a = k/6 (since the third difference of n³ alone is 3!=6). Here a = 6/6 = 1. Substituting n=1,2,3,4 into the known terms and solving the resulting system of equations gives b=1, c=−1, d=1, so the nth term is n³+n²−n+1. Checking: n=1 gives 1+1−1+1=2 ✓; n=4 gives 64+16−4+1=77 ✓; n=5 gives 125+25−5+1=146 ✓, matching every given term exactly. Cubic sequences like this appear whenever a quantity depends on a volume-like scaling relationship, such as the total number of unit cubes in a growing 3D stepped pyramid.

What's the difference between Recursive and Explicit (Closed-Form) Definitions?

Sequences can be defined two fundamentally different ways. An explicit (or closed-form) definition gives the nth term directly as a function of n alone — such as aₙ = 3n+2 — letting you jump straight to, say, the 100th term without knowing any earlier terms. A recursive definition instead gives each term in relation to the one(s) before it, such as aₙ = aₙ₋₁ + 5 with a₁=3, or the Fibonacci rule aₙ = aₙ₋₁ + aₙ₋₂. Recursive definitions often mirror how a sequence is actually generated, but computing a distant term requires working through every earlier term first, unless an explicit formula can be derived (as with arithmetic and geometric sequences). The Fibonacci sequence does have a known closed form (Binet's formula, involving the golden ratio raised to the power n), but it's far less obvious or memorable than the recursive rule, which is why Fibonacci numbers are almost always introduced and computed recursively in practice.

What's the difference between a Sequence and a Series (Sigma Notation)?

A sequence and a series are often confused but describe different things: a sequence is an ordered list of terms (3, 7, 11, 15, ...), while a series is the sum of those terms (3+7+11+15+...). Summation (sigma) notation, Σ, is the standard shorthand for writing a series compactly: Σ(k=1 to 5) of (4k−1) means 'substitute k=1,2,3,4,5 into 4k−1 and add the results.' Expanding: (4×1−1)+(4×2−1)+(4×3−1)+(4×4−1)+(4×5−1) = 3+7+11+15+19 = 55. This matches the arithmetic series sum formula directly: a=3, d=4, n=5, so S5 = 5/2×(2×3+4×4) = 2.5×22 = 55 ✓. Recognising sigma notation, and being able to expand it into an ordinary sum before applying the arithmetic or geometric series formulas, is a core skill bridging pattern-spotting in sequences with the more abstract summation notation used throughout further mathematics, statistics, and calculus. This connection also works in reverse: when a series is given in sigma notation but you're unsure whether it's arithmetic or geometric, substituting the first few values of k quickly reveals the underlying pattern — a common difference means use the arithmetic sum formula, a common ratio means use the geometric sum formula instead, exactly as if the terms had been listed out individually from the start.

What should I know about Sequences That Don't Fit a Simple Pattern?

Not every sequence settles into a recognisable arithmetic, geometric, or polynomial pattern — some diverge chaotically or are defined by rules (like the Collatz conjecture's 'if even halve it, if odd triple and add one') whose long-term behaviour remains an open mathematical question, a reminder that pattern-spotting techniques work brilliantly for the structured sequences typically encountered in coursework but don't automatically generalise to every possible rule-based sequence.