Ch 5 · 2.7–2.8 Sequences and Proportion · IGCSE Mathematics 0580
Topic 5 of 9 in Algebra and Graphs — IGCSE Mathematics 0580 — 56 lessons.
Introduction
Patterns of tiles, savings that grow each month, the number of handshakes in a group: sequences describe things that change step by step. Proportion describes how one quantity depends on another, like the stopping distance of a car (proportional to the square of its speed). (Syllabus sections 2.7 and 2.8.)
Key ideas — Term-to-term and position-to-term
A term-to-term rule says how to get the next term ("add 3"). The nth term gives any term directly from its position n. Subscript notation: T_n is the nth term of sequence T.
🧑🏫 Teacher's explanation
Imagine you are standing on a staircase. A term-to-term rule is like the instruction for each step: 'take one step up' or 'take two steps up'. It tells you how to get from where you are right now to the very next step. However, if you wanted to know how high you would be on the 100th step, counting every single step one by one would take forever.
📖 Definitions
🔍 Worked example — Easy
A sequence starts 5, 8, 11, 14, dots. Find the term-to-term rule and the first 4 terms.
🔍 Worked example — Standard
Find the nth term (T_n) for the sequence 7, 12, 17, 22, dots.
🔍 Worked example — Exam-style
The nth term of a sequence is given by T_n = n^2 + 3. Find the 10th term.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
(a) Find the first three terms of the sequence. (b) Determine whether 44 is a term in this sequence. Justify your answer. [3 marks]
Key ideas — Linear sequences
A constant difference d gives nth term dn + c: for 5, 8, 11, 14, dots, d = 3, so 3n + 2 (check: n = 1 gives 5).
🧑🏫 Teacher's explanation
Think of a linear sequence like a staircase where every step you take is exactly the same height. If you start at a certain height and each step adds a fixed amount, you are moving in a straight line. In mathematics, we call this a 'linear sequence' because if you plotted these points on a graph, they would form a perfectly straight line.
📖 Definitions
🔍 Worked example — Easy
Find the n^th term for the sequence: 7, 11, 15, 19, dots
🔍 Worked example — Standard
A sequence has the n^th term 5n - 2. Find the 50^th term.
🔍 Worked example — Exam-style
A sequence is given by T_n = 7 - 3n. Find the first three terms of this sequence.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Quadratic sequences
If the second differences are constant, the nth term contains an^2 with a = tfrac12 × second difference. Subtract an^2 from each term and find the linear part.
🧑🏫 Teacher's explanation
Think of a linear sequence like a staircase where every step is the same height. The 'first difference' is constant. A quadratic sequence is like a staircase where the height of each step changes at a steady rate. Imagine you are walking up a hill; the slope isn't constant, but the rate at which the slope increases is. In maths, we detect this by looking at the 'differences of the differences'.
📖 Definitions
🔍 Worked example — Exam-style
A quadratic sequence has the nth term an^2 + bn + c. The first three terms are 10, 19, 30. Find the 10th term.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
(a) Find the nth term of this sequence. (b) Find the 20th term. [4 marks]
Key ideas — Cubic, exponential and combined sequences
🧑🏫 Teacher's explanation
Think of sequences as a set of instructions for a growing pattern. A linear sequence is like walking up stairs (adding the same amount each time), but a cubic sequence is like building a solid cube of blocks; every time you increase the side length, the number of blocks grows much faster because you are adding volume in three dimensions. \\n\n\nExponential sequences are different; they don't grow by adding, they grow by multiplying. Imagine a single cell splitting into two, then four, then eight. The numbers explode in size very quickly. Sometimes, a sequence is a 'hybrid'—a mixture of two different rules, like a cubic pattern that also has a little bit of extra growth added to it every step. To solve these, we compare the 'messy' sequence to a 'pure' sequence we already know.
📖 Definitions
🔍 Worked example — Easy
Find the n^th term of the sequence: 2, 9, 28, 65, dots
🔍 Worked example — Standard
Find the n^th term of the exponential sequence: 3, 6, 12, 24, dots
🔍 Worked example — Exam-style
A sequence is defined by the rule 2^n + n. Find the 5^th term.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Direct and inverse proportion
Method: write the equation with k, substitute the given pair to find k, write the full formula, then use it.
🧑🏫 Teacher's explanation
Imagine you are buying apples. If one apple costs 50 cents, two cost 100 cents, and three cost 150 cents, the total cost is 'directly proportional' to the number of apples. As one goes up, the other goes up at a constant rate. We use a magic constant, k, to represent this rate. If the cost was y = kx, then k is just the price per apple.
📖 Definitions
🔍 Worked example — Easy
y is directly proportional to x. When x = 5, y = 20. Find y when x = 12.
🔍 Worked example — Standard
y is inversely proportional to x^2. When x = 3, y = 4. Find y when x = 2.
🔍 Worked example — Exam-style
y is inversely proportional to √(x). When x = 16, y = 3. Find the value of x when y = 1.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Worked examples
Is 100 a term of 5, 8, 11, dots? 3n + 2 = 100 ⇒ n = 32.7, not a whole number, so no.
🔍 Worked example — Easy
Find the 50th term of the sequence 7, 11, 15, dots and determine if 203 is a term in this sequence.
🔍 Worked example — Standard
y is directly proportional to x^2. When x = 4, y = 36. Find y when x = 10.
🔍 Worked example — Exam-style
y is inversely proportional to √(x). When x = 16, y = 3. Find x when y = 12.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) 4n + 3. (2) second difference 4, so 2n^2; 3 - 2, 9 - 8, 19 - 18, 33 - 32 = 1: 2n^2 + 1. (3) 162, 486; 2 × 3^n-1. (4) k = 20, p = 20/25 = 0.8. (5) k = 24, t = 8 hours.