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QUESTION ONE 12 marks Start a NEW booklet

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QUESTION ONE 12 marks Start a NEW booklet

    South Western Sydney Region HSC Mathematics Extension Study Day

    University of Western Sydney Campbelltown, 22nd September 2009

    THE BINOMIAL THEOREM

    Robert Yen, Hurlstone Agricultural High School

Outline

     1. Introduction 6. Finding the greatest coefficient

     2. Binomial expansions and Pascal’s triangle 7. Proving identities involving the sum of nn 3. C, a formula for Pascal’s triangle coefficients C kk

     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 that accompanies these notes can be found at the HSC Online

    website: http://hsc.csu.edu.au/maths n? This topic examines the general pattern for expanding (a + x)

    ? 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

    1(a + x) = a + x 2 1 1 2 22 (a + x)= a + 2ax + x3 1 2 1 3 3223(a + x)= a + 3ax + 3ax + x 4 1 3 3 1 4 432234(a + x)= a + 4ax + 6ax + 4ax + x 5 1 4 6 4 1 5 54322345 (a + x)= a + 5ax + 10ax + 10ax + 5ax + x6 1 5 10 10 5 1

     n The expansion of (a + x) has n + 1 terms, with the powers of a decreasing from n to 0 and the powers of x increasing from 0 to n. The sum of the powers in each term is always n. The coefficients of the terms appear in Pascal’s triangle, where each number is the sum of the two

    numbers above it.

     n3. C, A FORMULA FOR PASCAL’S TRIANGLE kn? Cfrom the Permutations and combinations topic also gives the value of row n, term k of k

    Pascal’s triangle, if the top of the triangle is row 0 and the first term in each row is term 0

     0 1 C 011 1 1 CC 0 1 222 1 2 1 CCC 0 1 2 3333 1 3 3 1 CCCC 0 1 2 3 44444 1 4 6 4 1 CCCCC 0 1 2 3 4 555555 1 5 10 10 5 1 CCCCCC 0 1 2 3 4 5 6666666 1 6 15 20 15 6 1 CCCCCCC 0 1 2 3 4 5 6 77777777 1 7 21 35 35 21 7 1 CCCCCCCC 0 1 2 3 4 5 6 7 888888888 1 8 28 56 70 56 28 8 1 CCCCCCCCC0 1 2 3 4 5 6 7 8

    n??n? C stands for coefficient as well as combination, and C is also written as ??k ??k??5? There are 3 ways of calculating C: 3

    5?4?35?45(a) Mentally: C = = 10 ?33?2?12

    n!5!1205n(b) Formula: C= = = 10, using C = 3 kk!(n?k)!3!2!6?2

    5?4?35?4?32?15!5??? This works for C because . 33?2?13?2?12?13!?2!

    n(c) Calculator key: pressing 5 C 3 = gives 10. r

The binomial theorem

     nnnnn-1 nn-2 2nn-3 3 nn-4 4nn(a + x) = Ca + Cax + Cax + Cax +Cax + … + Cx 0 1 2 3 4 n

     or in sigma notation:

     nCis the coefficient k nnnn?kkin the term that Cax (a + x) = ?kkcontains x in the k?0

    expansion

     the sum of terms the general term

     from k = 0 to n

     nProperties of C knn1. C=C = 1 First and last coefficients are 1 0 nnn2. C=C = n Second and second-last coefficients are n 1 n-1nn663. C = C Pascal’s triangle is symmetrical, for example, C = C kn-k2 4n+1nn4. C= C + C Pascal’s triangle result: each coefficient is the sum of the two k k-1k

    coefficients in the row above it

Example 1

     Use the binomial theorem to expand: Answers

     55432 (a) (a + 3) a + 15a + 90a + 270a + 405a + 243

     4432234 (b) (2x y) 16x 32xy + 24xy 8xy + y

    The binomial theorem: Robert Yen (page 2)

4. THE BINOMIAL THEOREM IN THE PAST 10 HSC EXAMS

    1999 last tested in Q7(b) Q3(b)

    1988, Q6(b) 2000 Q2(b)

    2001 Q2(d) Q5(b) Q5(c)

    2002 Q7(b) Q4(a)

    2003 Q2(d) Q3(c), Q8(a) Ext 2

    2004 Q7(b) Q4(c)

    2005 Q2(b) Q6(a)

    2006 Q2(b) Q6(b)

    2007 Q6(a)(i) Ext 2 Q4(a)

    2008 Q1(d), Q6(c)(i) Q6(c)(ii)

5. FINDING A PARTICULAR TERM

    Example 2 (2008 HSC, Question 1(d), 2 marks) 8412 Find an expression for the coefficient of xy in the expansion of (2x + 3y). 1284[Answer: C (2)(3)] 4

Steps for finding a particular term

1. Write a formula for the general term T of the expansion and simplify the formula, k1212-kk for example, T = C (2x) (3y). kk

    2. To find the term with the required power of x, solve an equation for k,

     for example, 12 k = 8, or k = 4.

     k must be a whole number or you have made a mistake.

3. Substitute the value of k into the T formula to find the required term. k

     thT is not the k term knk? In the expansion of (a + x), T is the term that contains x kththth? It is not the k term but actually the (k + 1) term, for example, T is the 4 term (TT, T30, 12, 3T), the one that contains x 3

    ? It is simpler to write out the first few terms of the expansion rather than try to memorise the

    sigma notation th? It is also better to avoid referring to the ‘k term’ and calling its formula T (as some k+1

    textbooks do) because students can get confused about the value of k to substitute (in

    Example 2 above, some substituted k = 5 ‘for the 5th term’ instead of k = 4) 8? Anyway, HSC questions tend to ask you to find, for example, ‘the term that contains x

    rather than ‘the 9th term’

Example 3 (2005 HSC, Question 2(b), 3 marks)

    121?? Use the binomial theorem to find the term independent of x in the expansion of 2x?. ??2x??

    [Answer: 126 720]

    The binomial theorem: Robert Yen (page 3)

Common student mistakes

    ? Giving the position of the term (‘the 5th term’) rather than the actual term

    ? Poor use of algebra, index laws, brackets and negative signs

    ? Wasting time expanding out all the terms

    ? Substituting wrong value for k, such as k + 1 instead 121284? In Example 2, giving the coefficient as C only instead of C (2)(3) 44

6. FINDING THE GREATEST COEFFICIENT 823 456 78 In (1 + 2x) = 1 + 16x + 112x + 448x+ 1120x + 1792x + 1792x+ 1024x + 256x, the

    greatest coefficient is 1792 (occurring twice).

     The term with the greatest coefficient usually occurs in the middle of an expansion because

    with the rows in Pascal’s triangle, the larger numbers are in the middle. In any expansion of n(a + x), the coefficients usually increase, reach a maximum, then decrease.

Example 4

    8k8tx Suppose (1 + 2x) = . k?k0?k8k (a) Find an expression for t, the coefficient of x. [Answer: C 2] kk

    28?k??tk?1 (b) Show that . ?tk?1k

     (c) Show that the greatest coefficient is 1792.

Steps for finding the greatest coefficient

    1. Write formulas for the general coefficient t and the next coefficient t. kk+1

    bnk()?t?1k2. Simplify to an expression of the form . ak?1t??k

    ?1n??n?1!c1 Note that = n and = c = c. ncn

    t?1k3. Solve > 1 to find the highest integer value of k. tk

    4. Find the value of t, the greatest coefficient. k+1

HOMEWORK EXERCISE (1988 HSC, Question 6(b), 6 marks)

    25k25tx Suppose (7 + 3x) = . k?k0?

     (i) Use the binomial theorem to write an expression for t, 0 ? k ? 25. k

    t3(25?k)?1k (ii) Show that . ?t7(k?1)k

     (iii) Hence or otherwise find the largest coefficient t. k

    25??cd You may leave your answer in the form. ??73??k??

    25??24187[Answer: (? 1.71 × 10)] ??t?737??7??

    The binomial theorem: Robert Yen (page 4)

    n7. PROVING IDENTITIES INVOLVING THE SUM OF COEFFICIENTS C k

    01) 1 (C) 0 1 (21 0121 1 1 2 (2) 2 (C) 1241 2 1 2 4 (2) 6 (C) 2361 3 3 1 3 8 (2) 20 (C) 3481 4 6 4 1 4 16 (2) 70 (C) 45101 5 10 10 5 1 5 32 (2) 252 (C) 56121 6 15 20 15 6 1 6 64 (2) 924 (C) 67141 7 21 35 35 21 7 1 7 128 (2) 3432 (C) 78161 8 28 56 70 56 28 8 1 8 256 (2) 12 870 (C) 8

Two important identities:

    nnnC?21. k?k?0

    555555555CCCCCCC??????????????15101051322 For example, . 012345?k0k?

    2nnn22. CC????knk?0

    4222222444444CCCCCC????? For example, . k01234?????????????k0?

    222228???????1464170C 4

     nIdentities involving the sum of coefficients can be proved by expanding (1 ? x) and then:

    ? substituting x = 0, 1 or -1, or

    ? equating coefficients, or

    ? differentiating or integrating.

    n The binomial theorem for (1 + x)

     nnnn2n3 n4nn (1 + x) = C + Cx + Cx + Cx +Cx + … + Cx 01 2 3 4 n

    nnnkCx or in sigma notation: (1 + x) = ?kk?0

Example 5

    nnnnC?2 Expand (1 + x) and substitute an appropriate value of x to prove that . k?k?0

    Example 6 2nnnn By considering that (1 + x) = (1 + x)(1 + x) and examining the coefficient of x on each side,

    n22nn??C?Cprove that . ?kn0k?

    The binomial theorem: Robert Yen (page 5)

    Hints for proving identities (by John Dillon, Head Teacher of Maths, Hurlstone AHS) n nnn2n3nn(1 + x)= C+ Cx + Cx + Cx + …+ C x 0 1 2 3 n nC’s with no x’s substituting a simple value such as x = 0 or x = 1 knC’s with alternating + and signs substituting a negative value for x knpowers of a number (say a) as well as C’s substituting x = a knC’s multiplied by k’s differentiating both sides knC’s divided by (k + 1)’s integrating both sides but don’t forget ‘+ c’ k

HOMEWORK EXERCISES nn+11. Expand both sides of the identity (1 + x)(1 + x) = (1 + x) and compare coefficients to prove n+1nnPascal’s triangle result C= C + C. k k-1knnn?1nkCn???22. Expand (1 + x) and differentiate both sides to prove that . k?k?1

    Example 7 (2006 HSC, Question 2(b), 2 marks) n (i) By applying the binomial theorem to (1 + x) and differentiating, show that 1

    nnnn????????n?1rn??11 nxxrxnx???????12.......??????????rn12????????

     (ii) Hence deduce that 1

    nnn??????nrn???111 nrn?????3...2...2.??????rn1??????

Example 8 (2008 HSC, Question 6(c), 5 marks)

     Let p and q be positive integers with p ? q.

     p + q (i) Use the binomial theorem to expand (1 + x), and hence write down the term 2

    pq?pq?1?x???? of which is independent of x. [Answer to (i) and (ii): ] ??qqx??

    pq?q1?x??1p?? (ii) Given that ???11x, apply the binomial theorem and the result of 3 ????qxx??

    pqpqpq???????????? part (i) to find a simpler expression for 1 + . ???????????????1122pp????????????

Common student mistakes

    ? Messy and careless working, unclear notation; not enough working, ‘fudging’ the answer nnn? Forgetting the first term C or the last term C x 0n

    ? Using series formulas or mathematical induction instead of the binomial theorem: this

    usually doesn’t work

    ? Not realising that the parts of the question are related

    ? Getting lost in sigma notation; from the examiners’ notes on the 2008 HSC exam (p.6):

    ‘Responses that used sigma notation were sometimes less successful than (students) who wrote

    out the sum showing at least three correct terms. Many ... misinterpreted this part of the

    question by stating which term was independent of x rather than by giving the independent

    term or, by being careless in their notation, failed to gain this mark.’

    ? If integrating, forgetting the constant at the end

    ? Starting a proof using the identity to be proved, rather than prove that LHS = RHS

    The binomial theorem: Robert Yen (page 6)

    Example 9 (2002 HSC, Question 7(b), 6 marks, HARD!) knn The coefficient of x in (1 + x), where n is a positive integer, is denoted by c (so c= C). kk k

     n-1 (i) Show that c + 2c + 3c + … + (n + 1)c = (n + 2) 2. 3 012n

    ccccnn012???...?(?1) (ii) Find the sum . 3 1?22?33?4(n?1)(n?2)

    1 Write your answer as a simple expression in terms of n. [Answer:] n?2

8. BINOMIAL PROBABILITY (for you to study at home)

     With binomial probability, we are concerned with repeated trials in which there are only two

    possible outcomes: we can call one outcome a success, with probability p, and the other outcome a failure, with probability q = 1 p. Examples of such outcomes are heads vs. tails, win vs. lose, true

    vs. false, boy vs. girl, defective vs. working.

    If a binomial trial is repeated n times, then the probability of r successes is nrn-rP(X = r) = C p q r

    X is called the random variable and its value ranges from 0 to n.

Example 10 (2007 HSC, Question 4(a), 4 marks)

     In a large city, 10% of the population has green eyes.

     (i) What is the probability that two randomly chosen people both have green eyes? 1

    [Answer: 0.01]

     (ii) What is the probability that exactly two of a group of 20 randomly chosen people 1

     have green eyes? Give your answer correct to three decimal places.

    [Answer: 0.285]

     (iii) What is the probability that more than two of a group of 20 randomly chosen people 2

     have green eyes? Give your answer correct to two decimal places.

    [Answer: 0.32]

Note that this is an application of the binomial theorem, because:

     P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + ... + P(X = 20) 200202011920218 2031720200 = C0.1 0.9 + C0.1 0.9 + C0.1 0.9+ C0.1 0.9 + …+ C0.10.9 0 1 2 3 20 20n = (0.9 + 0.1) that is, (q + p) Each probability is a 20 = 1 term of the expansion = 1. nThe sum of the of (q + p) probabilities of all possible events is 1 Common student mistakes

    ? Not using the complementary result as a shortcut

    ? Forgetting to include P(X = 0)

    ? Does ‘more than two’ include two?

    The binomial theorem: Robert Yen (page 7)

HOMEWORK EXERCISE (2004 HSC, Question 4(c))

     Katie is one of ten members of a social club. Each week one member is selected at random to

    win a prize.

     (i) What is the probability that in the first 7 weeks Katie will win at least 1 prize? 1

    79??[Answer: 1 ] ?0.5217??10??

     (ii) Show that in the first 20 weeks Katie has a greater chance of winning exactly 2 prizes 2

     than of winning exactly 1 prize.

     [Answer: P(X = 2) ? 0.2852 > P(X = 1) ? 0.2702]

    (iii) For how many weeks must Katie participate in the prize drawing so that she has 2

     a greater chance of winning exactly 3 prizes than of winning exactly 2 prizes?

     [Answer: 30 weeks]

9. HOW TO STUDY FOR MATHS: A 4-STEP APPROACH (P-R-A-C)

1. PRACTISE YOUR MATHS

    ? Master your skills, strengthen your ability

    ? Achieve a high level of understanding

2. REWRITE YOUR MATHS

    ? Summarise the theory and examples in your own words

    ? Work through all topics to see the big picture

    ? Achieve an overview of the whole course

3. ATTACK YOUR MATHS

    ? Identify your areas of weakness and work on overcoming them

    ? Fill in any gaps in your mathematical knowledge

4. CHECK YOUR MATHS

    ? Revise your understanding on mixed revision exercises and past HSC exams

Before an exam

    ? Review and memorise your topic summaries

    ? Practise on your weak areas

    ? Practise on HSC-style questions

    ? Anticipate the exam: the format and structure, the style of questions, planning your time

    during the exam.

Useful resources

    ? The NSW HSC Online website has tips, tutorials and links: http://hsc.csu.edu.au/maths ? The Board of Studies website has past HSC exams and Notes from the Marking Centre

    (examiners’ reports): www.boardofstudies.nsw.edu.au/hsc_exams

    ? The Mathematical Association of NSW sells booklets of past HSC exams with worked

    solutions; you may be able to buy these through your school: www.mansw.nsw.edu.au

    The binomial theorem: Robert Yen (page 8)

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