Binomial Theorem

# Binomial Theorem - Binomial Theorem Advanced Level Pure...

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Binomial Theorem Advanced Level Pure Mathematics Advanced Level Pure Mathematics Algebra Chapter 5 Binomial Theorem Appendix 1 Summation Sign and Product Sign 2 5.1 Introduction 6 5.3 Binomial Theorem for Positive Integral Index 10 5.4 The Greatest Coefficient and The Term with The Greatest Absolute Value 14 5.5 More Properties of Binomial Coefficients 16 5.6* Binomial Series 23 Prepared by K. F. Ngai Page 1 5

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Binomial Theorem Advanced Level Pure Mathematics Appendix 1 Summation Sign and Product Sign 1. The sigma notation : a r r n = 1 = a 1 + a 2 + + a n - 1 + a n Example a r r = 1 5 = a 1 + a 2 + a 3 + a 4 + a 5 ; r r 3 6 9 = = 2. a a a k k n l l n i i n = = = = = 0 0 0 3. ( ) a b a b k k k n k k n k k n + = + = = = 1 1 1 4. If c is a constant , then ( ) ca k k n = 1 = 5. If c is a constant , then c k n = 1 = 6. ( ) a a k k k n - - = 1 1 = 7. a a k k n k k n = + = - = 1 1 0 1 = 8. a a a i i n i j j n i n = = = = 1 2 1 1 9. The product notation : a k k n = 1 = a 1 a 2 a n - 1 a n Example 3 2 1 5 k k = = ; 3 1 k n = = 10. ( ) a b a b k k k n k k n k k n = = = = 1 1 1 Prepared by K. F. Ngai Page 2
Binomial Theorem Advanced Level Pure Mathematics 11. ( ) ca k k n = 1 = 12. a a k k k n - = 1 1 = 13*. r n n r n = = + 1 1 2 ( ) 14*. r n n n r n 2 1 1 2 1 6 = = + + ( )( ) 15*. r n n r r n r n 3 1 2 2 1 2 1 4 = = = + = ( ) 16. a k k n = 0 = a 0 + a k k n = 1 = a n + 17. kr lr k k n l l n = = = 0 0 18. ar ar ar k k n k k n k k n = + = - - = + = = 1 1 0 1 1 2 1 19. kp p k k n k - = - = 1 3 1 ? ? ? Prepared by K. F. Ngai Page 3

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Binomial Theorem Advanced Level Pure Mathematics 20. C x C x k n k k n k + - = - = 1 1 1 3 ? ? ? ? Example 1 Show that = + = = + n 1 r 1 n 2 k k 1 1 k 1 . Hence express = = + - n 1 k n 1 k 1 k 1 k 1 in term of n . Example 2 (a) Find the constants A and B in the following identity x B 1 x A x x 3 x 2 + + + + (b) Hence evaluate = + + - - n 2 k 2 n k k 3 k 1 k 1 lim . Prepared by K. F. Ngai Page 4
Binomial Theorem Advanced Level Pure Mathematics 5.1 Introduction 1. Binomial Theorem : (1 + x ) n = C C x C x C x n n n n n n 0 1 2 2 + + + + ... . 2. The term C r n x r is called the general term . Prepared by K. F. Ngai Page 5

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Binomial Theorem Advanced Level Pure Mathematics 3. C r n is called the binomial coefficients . 4. (1 - x ) n = C C x C x C x n n n n n n n 0 1 2 2 1 - + - + - ... ( ) 5. Factorial Notation : n ! = n ( n - 1)( n - 2) ⋅⋅⋅ 3 2 1 Example 3! = 3 × 2 × 1 ; 9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 N.B. (1) 0! = 1 (2) n ! = n ( n - 1)! (3) n ! = ( )! n n + + 1 1 Combinations : n r C , C r n , n r C Definition Let n be a positive integer and r non-negative integer with r n . The symbol C r n is defined as C P r n r n r n n n r r r n r n = = - = - - + !
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