Ch 2 · 12 Series · O Level Additional Mathematics 4037
Topic 2 of 3 in Series, Vectors and Counting — O Level Additional Mathematics 4037 — 58 lessons.
Introduction
A savings plan that adds the same amount each month is an arithmetic progression; a salary that rises by 5% a year is a geometric one. The binomial theorem multiplies out powers of brackets in one step, and its coefficients are the combinations from topic 11. (Syllabus topic 12.)
Key ideas — The binomial theorem
For a positive integer n (formula given):
🧑🏫 Teacher's explanation
Imagine you are packing a suitcase. If you have n identical slots and you want to pick r items of type b, the number of ways to do that is binomnr. The Binomial Theorem is just a way of systematically listing every possible combination of a and b that can occur when you multiply (a+b) by itself n times.
📖 Definitions
🔍 Worked example — Easy
Expand (x + 3)^4 fully.
🔍 Worked example — Standard
Find the coefficient of x^2 in the expansion of (2 - x/2 )^5.
🔍 Worked example — Exam-style
Find the term independent of x in the expansion of (x^2 - 3/x )^9.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — The general term
The term in b^r is dbinomnra^n-rb^r, for 0 ≤ r ≤ n. To find the term independent of x (the constant term), write the general term, collect the power of x, and set it to 0.
🧑🏫 Teacher's explanation
Imagine you are building a complex Lego tower using different colored bricks. If you want to know exactly how many ways you can pick a specific combination of bricks to reach a certain height, you need a mathematical 'recipe'. In the Binomial Theorem, that recipe is the general term. Instead of expanding a massive expression like (a+b)^10 by hand—which would take forever—the general term allows you to jump straight to any specific part of the expansion.
📖 Definitions
🔍 Worked example — Easy
Find the coefficient of x^2 in the expansion of (1 + 3x)^4.
🔍 Worked example — Standard
Find the term independent of x in the expansion of (x + 2/x)^6.
🔍 Worked example — Exam-style
In the expansion of (2 - x/2)^8, find the coefficient of the term containing x^4.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Arithmetic progressions
Common difference d; first term a:
🧑🏫 Teacher's explanation
Imagine you are climbing a staircase where every step is exactly the same height. If the first step is 15 ext cm and every subsequent step adds 15 ext cm, your height increases by a constant amount each time. This predictable, steady increase is what we call an Arithmetic Progression (AP). The 'common difference', denoted as d, is that constant step height.
📖 Definitions
🔍 Worked example — Easy
Find the 20^th term of the arithmetic progression: 5, 8, 11, dots
🔍 Worked example — Standard
The 3^rd term of an AP is 10 and the 8^th term is 25. Find the sum of the first 10 terms.
🔍 Worked example — Exam-style
Find the number of terms in the arithmetic progression: 7, 11, 15, dots, 103.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
(a) Find the value of x. (b) Find the sum of the first 20 terms of the progression. [5 marks]
Key ideas — Geometric progressions
Common ratio r; first term a:
🧑🏫 Teacher's explanation
Imagine you are walking towards a wall. In an Arithmetic Progression (AP), you take steps of a constant size, say 2 meters every time. You are adding a fixed amount. In a Geometric Progression (GP), you take a step that is half the size of the previous one. You are multiplying by a fixed amount.
📖 Definitions
🔍 Worked example — Easy
Find the 6th term of the geometric progression: 3, 6, 12, dots
🔍 Worked example — Standard
Calculate the sum of the first 8 terms of the GP: 10, 5, 2.5, dots
🔍 Worked example — Exam-style
A geometric series has a first term of 12 and a sum to infinity of 18. Find the common ratio r.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Sum to infinity
A GP converges only when -1 < r < 1; then
🧑🏫 Teacher's explanation
Imagine you are standing 2 metres away from a wall. You walk half the distance to the wall (1m), then half the remaining distance (0.5m), then half again (0.25m). Even though you keep moving forever, you will never actually pass the wall. The total distance you travel approaches a specific limit: 2 metres. This is what we call a 'sum to infinity'.
📖 Definitions
🔍 Worked example — Easy
Find the sum to infinity of the GP: 10, 5, 2.5, dots
🔍 Worked example — Standard
A GP has first term a = 12 and common ratio r = -1/3. Calculate the sum to infinity.
🔍 Worked example — Exam-style
Determine whether the series 5 + 10 + 20 + 40 + dots has a sum to infinity. Justify your answer.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples
Find the term independent of x in (x^2 + 1/x )^6. General term binom6r(x^2)^6-r (tfrac1x )^r = binom6rx^12 - 3r. Constant when r = 4: binom64 = 15.
🔍 Worked example — Easy
Find the term independent of x in the expansion of (x^2 - 3/x )^9.
🔍 Worked example — Standard
The first term of an AP is 7 and the 10^th term is 34. Find the sum of the first 10 terms.
🔍 Worked example — Exam-style
A geometric progression has first term 12 and common ratio 0.4. Find the sum to infinity.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) 243 - 810x + 1080x^2. (2) general term binom9r(2x)^9-r(-x^-2)^r, power 9 - 3r = 0 ⇒ r = 3: binom93 2^6(-1)^3 = -5376. (3) d = 4, a = 5: S_20 = 10(10 + 76) = 860. (4) S_∞ = 81/2/3 = 121.5; 121.5 (1 - (tfrac13 )^n ) > 121 ⇒ 3^n > 243 ⇒ n = 6. (5) r = -2, so |r| > 1: the terms grow in size and the sum does not converge.