Upload File

the binomial expansion - solutions

prev
next
out of 2
Download The Binomial Expansion - Solutions

Upload: wolfretonmaths

Post on 10-Oct-2015

30 views

Category:

Documents


0 download

Report

TRANSCRIPT

  • AQA Core 2 Sequences and series

    1 of 2 15/01/13 MEI

    Section 4: The binomial expansion Solutions to Exercise

    1. Pascals triangle: 1

    1 1

    1 2 1

    1 3 3 1

    1 4 6 4 1

    1 5 10 10 5 1

    1 6 15 20 15 6 1

    (i) 6 6 5 4 3 2( 1) 6 15 20 15 6 1x x x x x x x

    (ii) 5 5 4 1 3 2 2 3 4 5

    5 4 3 2

    5 4 3 2

    ( 2) 5 ( 2) 10 ( 2) 10 ( 2) 5 ( 2) ( 2)

    5 2 10 4 10 8 5 16 32

    10 40 80 80 32

    x x x x x x

    x x x x x

    x x x x x

    (iii) 4 4 3 2

    4 3 2

    4 3 2

    (2 1) (2 ) 4(2 ) 6(2 ) 4(2 ) 1

    16 4 8 6 4 8 1

    16 32 24 8 1

    x x x x x

    x x x x

    x x x x

    (iv) 3 3 2 2 3

    2 3

    2 3

    (2 3 ) 2 3(2) ( 3 ) 3(2)( 3 ) ( 3 )

    8 3 4 3 3 2 9 27

    8 33 54 27

    x x x x

    x x x

    x x x

    2. (i) 8 8 7 6

    3

    1 2 38 7 56

    (ii) 9 9 9 8

    5 4

    27 6

    1 2 3 49 2 7 126

    (iii) 12 12

    4

    11 10 59

    1 2

    3 411 9 5 495

    (iv) 20 20 20

    18 2

    1019

    1 2

    10 19 190

    3. (i) Term in x4

    4 4 415 15 14 13 12(2 ) 16 21840

    4 1 2 3 4x x x

    Coefficient of x4 is 21840

  • AQA C2 Series 4 Exercise solutions

    2 of 2 15/01/13 MEI

    (ii) Term in x23 2 23 25 2 232

    23 23

    25(3) ( ) (3) ( )

    23

    25 249 2700

    1 2

    x C x

    x x

    Coefficient of x23 is -2700.

    (iii) Term in x 4 6

    6 2

    4

    10 10 9 8 71210

    4 1 2 3 4

    xx xx x

    Coefficient of x2 is 210.

    4. 4 2(1 ) 1 4 6 ...x x x 7 2(1 ) 1 7 21 ...x x x

    4 7 2 2

    2 2

    2 2 2

    2

    (1 ) (1 ) 1 4 6 ... 1 7 21 ...

    (1 7 21 ) 4 (1 7 ) 6 ...

    1 7 21 4 28 6 ...

    1 3 ...

    x x x x x x

    x x x x x

    x x x x x

    x x

    5. (i) 15 2 215 14

    (1 ) 1 15 ... 1 15 105 ...1 2

    x x x x x

    (ii) Putting x = 0.01: 15 2

    15

    (1 0.01) 1 15 0.01 105 0.01 ...

    0.99 1 0.15 0.0105 ... 0.8605

    (iii) Percentage error 15

    15

    0.8605 0.99100 0.051%

    0.99

    6. (i) 9 2 3

    9 2 3212 2

    9 8 9 8 71 1 9 ...2 2 1 2 2 1 2 3 2

    1 9 ...

    x x x x

    x x x

    (ii) Putting x = 0.1: 9

    9 2 3212 2

    9

    0.11 1 0.1 9 0.1 0.1 ...2

    1.05 1 0.45 0.09 0.0105 ... 1.5505

    (iii) Percentage error 9

    9

    1.5505 1.05100 0.053%

    1.05


C2: Chapter 5 Binomial Expansion

Term 1 : Unit 3 Binomial Theorem 3.1 The Binomial Expansion of (1 + b) n 3.2 The Binomial Expansion of (a + b) n

Binomial expansion, power series, limits, approximations, …personal.maths.surrey.ac.uk/st/S.Gourley/series.pdf · The binomial expansion as discussed up to now is for the case when

Binomial expansion, power series, limits, approximations ...personal.ee.surrey.ac.uk/S.Gourley/series.pdfTaylor’s expansion, and the related Maclaurin expansion discussed below,

BINOMIAL EXPANSION. Binomial Expansions Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 The binomial theorem provides a useful method

Notes 32 Multiplying Polynomials - Garrett Academygarrett.ccsdschools.com/UserFiles/Servers/Server_2879574/File... · Binomial Expansion For a binomial expansion of the form (a +

Ch5 Binomial Expansion

- Plainfield North High Schoolpnhs.psd202.org/documents/TAcosta/1511817744.pdfThe Binomial Theorem states what we have observed about binomial expansion in the previous lesson: For

GARYMATHS.WEEBLYgarymaths.weebly.com/.../c4_study_pack_part_1-1_1.docx · Web viewPartial Fractions Basic topics/various/algebraic fractions. P32-47 Binomial expansion Standard topics/various/binomial

5) C2 the Binomial Expansion

Binomial Expansion And More

C2 Sequences & Series: Binomial Expansion ......2010/12/11  · C2 Sequences & Series: Binomial Expansion PhysicsAndMathsTutor.com Edexcel Internal Review 3 8. (a) Find the first 4

maccss.ncdpi.wikispaces.netmaccss.ncdpi.wikispaces.net/file/view/Algebra+Theme... · Web viewThe Binomial Theorem describes the algebraic expansion of powers of a binomial. There

C2 Sequences & Series: Binomial Expansion · C2 Sequences & Series: Binomial Expansion Edexcel Internal Review 2 5. (a) Find the first 4 terms, in ascending powers of x, of the binomial

NCERT Solutions for Class 11 Maths Chapter 8 NCERT Solutions for Class 11 Maths Chapter 8 Binomial Theorem Class 11 Chapter 8 Binomial Theorem Exercise 8.1, 8.2, miscellaneous Solutions

Expansion of Binomials. (x+y) n The expansion of a binomial follows a predictable pattern Learn the pattern and you can expand any binomial

Sequences and Series Part 1b; Binomial Expansion

UWS - Binomial Theorem · Determine the first 3 terms in the binomial expansion of ! . Hence, obtain an estimate for ! . 11. Using the Binomial Theorem, in ascending powers of x,

The Binomial Theorem. Expansion of Binomials Binomial Expansion Coefficients Pascal’s Triangle Row

Binomial Expansion Honors Advanced Algebra Presentation 2-3

Unit 6 – Bernoulli and Binomial Distributions Homework ...people.umass.edu/~biep540w/pdf/sol_bernoulli.pdfBernoulli and Binomial Distributions SOLUTIONS sol_bernoulli.docx Page 1

Www.le.ac.uk Binomial Expansion and Maclaurin series Department of Mathematics University of Leicester

Ch.3 Binomial Expansion - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C4/Topic-Qs... · In the binomial expansion of f x, in ascending powers of . x,

Holt McDougal Algebra 2 6-2 Multiplying Polynomials Multiply polynomials. Use binomial expansion to expand binomial expressions that are raised to positive

The Binomial Theorem - Arizona State Universityboerner/mat243/6.4 Binomial...The Binomial Theorem Let’s generalize this understanding. In the expansion + 𝑛,the coefficient of

Binomial Expansion MARK SCHEME

The coefficients in the binomial expansion of (1 + x ) 5

Chapter 8 Binomial Theorem - Ncert Help 11/maths/ncert solutions... · Therefore, the coefficient of xn in the expansion of (1 + x)2n is twice the coefficient of xn in the expansion

Binomial Coefficient. Definition of Binomial coefficient For nonnegative integers n and r with n > r the expansion (read “n above r”) is called a binomial

A-Level Unit Test: Series and Sequences Binomial Expansion 1b

Counting and the binomial expansion

Binomial Expansion Reflection

tECHNOLOGY PARTNERs, oem partners binomial solutions pvt. ltd

SSWLSR318CP18011509562 - · PDF fileYou can use the binomial theorem to expand any power of a binomial expansion. -nco(x Example 1: Use the Binomial Theorem a) Use the binomial theorem

¹©¹ 16 16 1 Binomial expansion Mixed exercise 4 §·§· §· x ...Binomial expansion Mixed exercise 4 1 a i (1 − 4x)3 Use binomial expansion with n = 3 and x = − 4x (3)(2)(