Lec2p2 - JUST THE MATHS UNIT NUMBER 2.2 SERIES 2(Binomial series by A.J.Hobson 2.2.1 2.2.2 2.2.3 2.2.4 Pascals Triangle Binomial Formulae Exercises

# Lec2p2 - JUST THE MATHS UNIT NUMBER 2.2 SERIES 2(Binomial...

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“JUST THE MATHS” UNIT NUMBER 2.2 SERIES 2 (Binomial series) by A.J.Hobson 2.2.1 Pascal’s Triangle 2.2.2 Binomial Formulae 2.2.3 Exercises 2.2.4 Answers to exercises
UNIT 2.2 - SERIES 2 - BINOMIAL SERIES INTRODUCTION In this section, we shall be concerned with the methods of expanding (multiplying out) an expression of the form ( A + B ) n , where A and B are either mathematical expressions or numerical values, and n is a given number which need not be a positive integer. However, we shall deal first with the case when n is a positive integer, since there is a useful aid to memory for obtaining the result. 2.2.1 PASCAL’S TRIANGLE Initially, we consider some simple illustrations obtainable from very elementary algebraic techniques in earlier work: 1. ( A + B ) 1 A + B. 2. ( A + B ) 2 A 2 + 2 AB + B 2 . 3. ( A + B ) 3 A 3 + 3 A 2 B + 3 AB 2 + B 3 . 4. ( A + B ) 4 A 4 + 4 A 3 B + 6 A 2 B 2 + 4 AB 3 + B 4 . OBSERVATIONS (i) We notice that, in each result, the expansion begins with the maximum possible power of A and ends with the maximum possible power of B . (ii) In the sequence of terms from beginning to end, the powers of A decrease in steps of 1 while the powers of B increase in steps of 1. 1
(iii) The coefficients in the illustrated expansions follow the diagramatic pattern calledPASCAL’S TRIANGLE:11121133114641and this suggests a general pattern where each line begins and ends with the number 1and each of the other numbers is the sum of the two numbers above it in the previousline. For example, the next line would be15101051giving the result5. (A+B)5A5+ 5A4B+ 10A3B2+ 10A2B3+ 5AB4+B5.(iv) The only difference which occurs in an expansion of the form(A-B)nis that the terms are alternately positive and negative. For instance,