Binomial theorem ppt

binomial distribution ppt presentation and binomial option pricing model ppt
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Binomials MATH 1510 Lili Shen The Binomial Theorem An expression of the form a+b is called a binomial. The purpose of this section is to find a formula that gives the expansion of n (a+b) + for all n2N and we will prove it using mathematical induction.n Expanding (a+b) MATH 1510 Lili Shen n To find a pattern in the expansion of(a+b) , we first look at The Binomial Theorem some special cases: 1 (a+b) = a+b; 2 2 2 (a+b) = a + 2ab+b ; 3 3 2 2 3 (a+b) = a + 3a b+ 3ab +b ; 4 4 3 2 2 3 4 (a+b) = a + 4a b+ 6a b + 4ab +b ; 5 5 4 3 2 2 3 4 5 (a+b) = a + 5a b+ 10a b + 10a b + 5ab +b ; . . .n Expanding (a+b) MATH 1510 We notice that the exponents of a decrease and the Lili Shen exponents of b increase: The Binomial 5 5 0 4 1 3 2 2 3 1 4 0 5 (a+b) = a b + 5a b + 10a b + 10a b + 5a b +a b : Theorem The coefficients in the expansions constitute a Pascal’s triangle: 0 (a+b) 1 1 (a+b) 1 1 2 (a+b) 1 2 1 3 (a+b) 1 3 3 1 4 (a+b) 1 4 6 4 1 5 (a+b) 1 5 10 10 5 1n Expanding (a+b) MATH 1510 Lili Shen The Binomial Theorem Example 7 Find the expansion of (a+b) using Pascal’s triangle.n Expanding (a+b) MATH 1510 Lili Shen Solution. The Binomial Theorem Find the seventh row of Pascal’s triangle: 5 (a+b) 1 5 10 10 5 1 6 (a+b) 1 6 15 20 15 6 1 7 (a+b) 1 7 21 35 35 21 7 1 Thus 7 7 6 5 2 4 3 3 4 2 5 6 7 (a+b) = a +7a b+21a b +35a b +35a b +21a b +7ab +b :The binomial coefficients MATH 1510 Lili Shen The Binomial Theorem Although Pascal’s triangle is useful in finding the binomial expansion for reasonably small values of n, it is not practical n for finding (a+b) for large values of n. The reason is that the method we use for finding the successive rows of Pascal’s triangle is recursive. Thus to find the 100th row of this triangle, we must first find the preceding 99 rows.n factorial MATH 1510 Lili Shen The Binomial Theorem The product of the first n natural numbers is denoted by n, called n factorial, i.e., n = 1 2 3(n 1)n: We also define 0 = 1 which makes many formulas involving factorials shorter and easier to write.The binomial coefficients MATH 1510 Lili Shen The Binomial Theorem Definition ‚ Œ n For n;r2N with r n, the binomial coefficient is r defined by ‚ Œ n n = : r r(nr)The binomial coefficients MATH 1510 Lili Shen The Binomial Theorem The binomial coefficients also arises in many areas of mathematics other than algebra, especially in combinatorics. ‚ Œ ‚ Œ n n often reads as “n choose r”, because there are r r ways to choose r elements, disregarding their order, from a set of n elements.The binomial coefficients MATH 1510 Lili Shen The Binomial Theorem Proposition ‚ Œ ‚ Œ n n n(n 1)(n 2) : : :(nr + 1) = = . r nr rThe binomial coefficients MATH 1510 Lili Shen The Binomial Theorem Example ‚ Œ ‚ Œ 9 9 9 8 7 6 (1) = = = 126. 4 5 4 3 2 1 ‚ Œ ‚ Œ 10 10 10 9 8 (2) = = = 120. 3 7 3 2 1The binomial coefficients MATH 1510 Lili Shen The Binomial Theorem To see the connection between the binomial coefficients n and the binomial expansion of (a+b) , note that ‚ Œ ‚ Œ ‚ Œ ‚ Œ ‚ Œ ‚ Œ 5 5 5 5 5 5 = 1; = 5; = 10; = 10; = 5; = 1: 0 1 2 3 4 5 These are precisely the entries in the fifth row of Pascal’s triangle.The binomial coefficients MATH 1510 In fact, Pascal’s triangle can be expressed as Lili Shen  The Binomial 0 Theorem 0   1 1 0 1    2 2 2 0 1 2     3 3 3 3 0 1 2 3      4 4 4 4 4 0 1 2 3 4       5 5 5 5 5 5 0 1 2 3 4 5                   n n n n n     0 n1 1 2 nThe binomial coefficients MATH 1510 Lili Shen The Binomial Theorem To demonstrate that this pattern holds, we need to show that any entry in this version of Pascal’s triangle is the sum of the two entries diagonally above it; that is, ‚ Œ ‚ Œ ‚ Œ n n n+ 1 + = r 1 r r for all n;r2N with r n.The binomial coefficients MATH 1510 Lili Shen Indeed, ‚ Œ The Binomial Theorem n+ 1 (n+ 1) = r r(n+ 1r) n(n+ 1r)+r = r(n+ 1r) n(n+ 1r) nr = + r(n+ 1r) r(n+ 1r) n n = + r(nr) (r 1)n(r 1) ‚ Œ ‚ Œ n n = + : r r 1The binomial theorem MATH 1510 Lili Shen The Binomial Theorem We are now ready to state the Binomial Theorem: Theorem ‚ Œ ‚ Œ n n X X n n n nr r r nr (a+b) = a b = a b . r r r=0 r=0The binomial theorem MATH 1510 Proof. Lili Shen ‚ Œ n X n nr r n The Binomial Let P(n) denote the statement (a+b) = a b . Theorem r r=0 ‚ Œ ‚ Œ 1 1 1 1 1 Step1. P(1) is true since (a+b) = a + b . 0 1 Step2. Suppose P(k) is true, i.e., ‚ Œ k X k k kr r (a+b) = a b : r r=0 We show that P(k + 1) is true, i.e., ‚ Œ k+1 X k + 1 k+1 k+1r r (a+b) = a b : r r=0The binomial theorem MATH 1510 Lili Shen Indeed, The Binomial k+1 k Theorem (a+b) = (a+b)(a+b) " ‚ Œ k X k kr r = (a+b) a b r r=0 " " ‚ Œ ‚ Œ k k X X k k kr r kr r = a a b +b a b r r r=0 r=0 ‚ Œ ‚ Œ k k X X k k k+1r r kr r+1 = a b + a b r r r=0 r=0 " ‚ Œ ‚ Œ k k1 X X k k k+1 k+1r r kr r+1 k+1 = a + a b + a b +b r r r=1 r=0The binomial theorem MATH 1510 Lili Shen " The Binomial ‚ Œ ‚ Œ k k X X Theorem k k k+1 k+1r r k+1r r k+1 = a + a b + a b +b r r 1 r=1 r=1 " –‚ Œ ‚ Œ™ k X k k k+1 k+1r r k+1 = a + + a b +b r r 1 r=1 " ‚ Œ k X k + 1 k+1 k+1r r k+1 = a + a b +b r r=1 ‚ Œ k+1 X k + 1 k+1r r = a b : r r=0 Thus P(k + 1) follows from P(k), completing the proof.The binomial theorem MATH 1510 Lili Shen The Binomial Theorem Example 4 Use the binomial theorem to expand (x +y) .

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