Ch 6 · 1.6 Series · A Level Mathematics 9709
Topic 6 of 8 in Pure Mathematics 1 — A Level Mathematics 9709 — 64 lessons.
Introduction
Saving the same extra amount each month gives an arithmetic pattern; money growing by the same percentage each year gives a geometric one. The binomial expansion multiplies out brackets such as (2 + 3x)^5 in one step. Paper 1 almost always has a question on one or both. (Syllabus section 1.6, Paper 1.)
Key ideas — Factorials and binomial coefficients
n! = n × (n-1) × dots × 2 × 1, with 0! = 1. The binomial coefficient is
🧑🏫 Teacher's explanation
Imagine you are organizing a small tournament. If you have 5 players and you want to pick 2 to play a final match, the order doesn't matter—picking Alice then Bob is the same as picking Bob then Alice. Factorials (n!) help us count how many ways we can arrange things, while binomial coefficients binomnr help us count how many ways we can choose a subset from a larger group.
📖 Definitions
🔍 Worked example — Easy
Calculate the value of binom73.
🔍 Worked example — Standard
Find the coefficient of x^2 in the expansion of (2x + 3)^5.
🔍 Worked example — Exam-style
Show that binomnr = binomnn-r for n=10, r=4.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — The binomial expansion
For a positive integer n:
🧑🏫 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 and the rest of type A, the number of ways to do this is binomnr. The binomial expansion is just a way of systematically listing every possible combination of these items.
📖 Definitions
🔍 Worked example — Easy
Expand (x + 3)^4 fully.
🔍 Worked example — Standard
Find the coefficient of x^2 in the expansion of (2 - 3x)^5.
🔍 Worked example — Exam-style
Find the coefficient of x^3 in the expansion of (1 + 2x)(3 - x)^5.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Arithmetic progressions (AP)
Each term is the previous one plus a common difference d: a,\ a + d,\ a + 2d, dots
Key ideas — Geometric progressions (GP)
Each term is the previous one times a common ratio r: a,\ ar,\ ar^2, dots
🧑🏫 Teacher's explanation
Imagine you are playing a video game where every time you level up, your score doubles. If you start with 10 points, your next score is 20, then 40, then 80. This is a Geometric Progression (GP). Unlike an Arithmetic Progression where you add the same amount every time, in a GP, you multiply by the same amount every time. This multiplier is called the 'common ratio', denoted as r.\n\nThink of it like a snowball rolling down a hill. If the snowball's volume increases by a fixed percentage every meter it rolls, the growth isn't a steady climb; it's an explosion of size. In maths, we call this 'exponential growth'. If that multiplier r is a fraction between -1 and 1, the snowball eventually stops growing and settles at a specific size—this is what we call 'convergence'.
📖 Definitions
🔍 Worked example — Easy
Find the 7th term of a GP where the first term a = 5 and the common ratio r = 3.
🔍 Worked example — Standard
The first three terms of a GP are x, x+2, and x+6. Find the value of x.
🔍 Worked example — Exam-style
A geometric series has a first term of 10 and a common ratio of 0.6. Find the sum to infinity.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Sum to infinity
If |r| < 1 (i.e. -1 < r < 1), r^n → 0 as n grows, so the GP converges and
🧑🏫 Teacher's explanation
Imagine you are standing 2 metres away from a wall. You walk half the distance to the wall (1m), then half of that remaining distance (0.5m), then half again (0.25m). Even though you keep moving forever, you will never actually pass the wall; you are simply getting infinitely close to it. This 'limit' is the sum to infinity.
📖 Definitions
🔍 Worked example — Easy
Find the sum to infinity of the geometric series: 10 + 5 + 2.5 + 1.25 + dots
🔍 Worked example — Standard
A geometric series has a first term of 12 and a sum to infinity of 18. Find the common ratio r.
🔍 Worked example — Exam-style
The first three terms of a geometric progression are x, x+2, x+6. Find the sum to infinity of the series.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples — Example 1: equating coefficients
The coefficient of x^2 in (1 - 4x)^6 is 12 times the coefficient of x^2 in (2 + ax)^5. Find the positive constant a.
🔍 Worked example — Easy
Find the coefficient of x^2 in the expansion of (1 + 3x)^4.
🔍 Worked example — Standard
The coefficient of x^2 in (1 - 4x)^6 is 12 times the coefficient of x^2 in (2 + ax)^5. Find the positive constant a.
🔍 Worked example — Exam-style
In the expansion of (k + x)^5, the coefficient of x^3 is 80. Find the value of k.
⚠️ Common errors
Worked examples — Example 2: a product
Find the coefficient of x^2 in (1 + 2x)(3 - x)^5.
🔍 Worked example — Easy
Find the coefficient of x in the expansion of (2 + 3x)^4.
🔍 Worked example — Standard
Find the coefficient of x^2 in the expansion of (1 + 2x)(3 - x)^5.
🔍 Worked example — Exam-style
Find the coefficient of x^3 in the expansion of (1 - x)^5 (1 + x)^2.
⚠️ Common errors
Worked examples — Example 3: an AP
The 3rd term of an AP is 11 and the 10th term is 32. Find the sum of the first 20 terms.
🔍 Worked example — Easy
Find the 15th term of an AP where the first term a = 4 and the common difference d = 5.
🔍 Worked example — Standard
The 3rd term of an AP is 11 and the 10th term is 32. Find the sum of the first 20 terms.
🔍 Worked example — Exam-style
Three numbers are in AP. The sum of the first and third term is 20, and the sum of the second and third term is 31. Find the common difference.
⚠️ Common errors
Worked examples — Example 4: a GP and its sum to infinity
A GP has first term 24 and sum to infinity 96. Find the common ratio and the sum of the first 5 terms.
🔍 Worked example — Easy
A GP has first term a = 10 and common ratio r = 0.5. Find the sum to infinity.
🔍 Worked example — Standard
A GP has first term 24 and sum to infinity 96. Find the common ratio r and the sum of the first 5 terms S_5.
🔍 Worked example — Exam-style
The first three terms of a GP are x, x+3, and x+9. Find the value of x and the common ratio r.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) 64 - 576x + 2160x^2. (2) binom63 x^3 (tfrac2x )^3 = 20 × 8 = 160. (3) S_n = tfrac n2(14 + 4(n-1)) = 2n^2 + 5n > 1000, so n = 22 (S_21 = 987, S_22 = 1078). (4) (k+4)^2 = (k-2)(3k+2) ⇒ k^2 + 8k + 16 = 3k^2 - 4k - 4 ⇒ 2k^2 - 12k - 20 = 0 ⇒ k^2 - 6k - 10 = 0 ⇒ k = 3 ± √(19). (5) ar = 12, ar^4 = -tfrac32 ⇒ r^3 = -tfrac18, r = -tfrac12, a = -24, S_∞ = -24/3/2 = -16.