# ups and downs

Essay by tantan21University, Bachelor'sB, July 2014

South Western Sydney Region HSC Mathematics Extension 1-2 Study Day

Bankstown Senior College, September 2012

THE BINOMIAL THEOREM

Robert Yen

Outline

1. Introduction 6. Finding the greatest coefficient

2. Binomial expansions and Pascal's triangle 7. Proving identities involving the sum of

3. nCk, a formula for Pascal's triangle coefficients nCk

4. The binomial theorem in the past 10 HSC exams 8. Binomial probability

5. Finding a particular term 9. How to study for Maths: a 4-step approach

1. INTRODUCTION

The PowerPoint presentation for these notes can be found at HSC Online: http://hsc.csu.edu.au/maths/ext1/binomial_theorem/

This topic examines the general pattern for expanding (a + x)n

It is a difficult topic because it involves new work on high-level algebra and is learned at the end of the course with little time for practice and revision

HSC questions involving this topic are often targeted at better Extension 1 students, especially when they appear in Question 7, so if you are aiming to achieve at the highest band (E4) in this course, work on mastering this topic to excel in the exam

There are no shortcuts to success in this topic: you just have to learn the theory to develop a full understanding

2. BINOMIAL EXPANSIONS AND PASCAL'S TRIANGLE

 Binomial expansion No. of terms Coefficients of terms (a + x)1 = a + x 2 1 1 (a + x)2 = a2 + 2ax + x2 3 1 2 1 (a + x)3 = a3 + 3a2x + 3ax2 + x3 4 1 3 3 1 (a + x)4 = a4 + 4a3x + 6a2x2 + 4ax3 + x4 5 1 4...