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Let fn be the n-th Fibonacci number of Example 3c in 8.1, k=n k=n Σ'=1+2+...+ B₁=k²= 1² +2²+. k=1 k=1 by definition fo = Ao = Bo = 0. (a) Give a recursive definition of the numbers fn, An, Bn with n20 +n²;

Let fn be the n-th Fibonacci number of Example 3c in 8.1, k=n k=n Σ'=1+2+...+ B₁=k²= 1² +2²+. k=1 k=1 by definition fo = Ao = Bo = 0. (a) Give a recursive definition of the numbers fn, An, Bn with n20 +n²;

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.3: Divisibility
Problem 49E: 49. a. The binomial coefficients are defined in Exercise of Section. Use induction on to prove...
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Let fn be the n-th Fibonacci number of Example 3c in 8.1. k=n k=n Σ'=1+2+...+8, B₁=k²= 1² +2²+. k=1 k=1 by definition fo = Ao = Bo = 0. (a) Give a recursive definition of the numbers fn, An, Bn with n ≥ 0 (d) Using only part (a), show that X X f(x)=x+xf(x) + x²f(x), A(x)=xA(x) + B(x) = xB(x) + (1-x)²¹ (1-x)² Hint: You'll need to use identities such as the following: 31 1 1 (--(---)-([^) - ¹2m(x-1)x³-²2, = = Σ (1-x)³ 1-2 n=0 n=0 + n²; + 22-2 (1-x)³
Transcribed Image Text:Let fn be the n-th Fibonacci number of Example 3c in 8.1. k=n k=n Σ'=1+2+...+8, B₁=k²= 1² +2²+. k=1 k=1 by definition fo = Ao = Bo = 0. (a) Give a recursive definition of the numbers fn, An, Bn with n ≥ 0 (d) Using only part (a), show that X X f(x)=x+xf(x) + x²f(x), A(x)=xA(x) + B(x) = xB(x) + (1-x)²¹ (1-x)² Hint: You'll need to use identities such as the following: 31 1 1 (--(---)-([^) - ¹2m(x-1)x³-²2, = = Σ (1-x)³ 1-2 n=0 n=0 + n²; + 22-2 (1-x)³
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