Suppose f : N→N satisestherecurrencerelation f(n + 1) (f(n) 2 if f(n)iseven 3f(n)+ 1 if f(n)isodd...

Suppose f : N→N satisestherecurrencerelation f(n + 1) (f(n) 2 if f(n)iseven 3f(n)+ 1 if f(n)isodd . Notethatwiththeinitialcondition f(0) 1,thevaluesofthefunction are: f(1) 4, f(2) 2, f(3) 1, f(4) 4, and so on, the images cyclingthroughthosethreenumbers. Thus f isNOTinjective(andalso certainlynotsurjective). Mightitbeunderotherinitialconditions?3 (a) If f satisestheinitialcondition f(0) 5,is f injective? Explain whyorgiveaspecicexampleoftwoelementsfromthedomain withthesameimage. (b) If f satisestheinitialcondition f(0) 3,is f injective? Explain whyorgiveaspecicexampleoftwoelementsfromthedomain withthesameimage. (c) If f satisestheinitialcondition f(0) 27,thenitturnsoutthat f(105) 10 and no two numbers less than 105 have the same image. Could f beinjective? Explain. (d) Prove that no matter what initial condition you choose, the functioncannotbesurjective.

Expert Solution

Colby Messinger answered 2 years ago

2. The Fibonacci sequence is defined as f(n) = f(n - 1) + f(n - 2)...

2. The Fibonacci sequence is defined as f(n) = f(n - 1) + f(n - 2) with f(0) = 0 and f(1) = 1. Find f(54) by a program or maually. Note that this number must be positive and f(53) = 53.......73 (starting with 53 and ending with 73). I must admit that my three machines including a desktop are unable to find f(54) and they quit during computation. The answer is f(54) = 86267571272 */ The Java code: public...

Fibonacci Sequence: F(0) = 1, F(1) = 2, F(n) = F(n − 1) + F(n −...

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Suppose f(1) = 2, g(1) = −1, f′(1) = 3, g′(1) = 2, f(2) = 1,...

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Define the following function f(n) = 5(2^n)-(2^(n-1)), n ≥ 1. Write a recursive definition for the...

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2. Let f(x) ≥ 0 on [1, 2] and suppose that f is integrable on [1,...

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If f(n) = 3n+2 and g(n) = n, then Prove that f(n) = O (g(n))

proof: L t^(n+1)f(t)=(-1)^(n+1)(d^(n+1)/ds^(n+1))*F(s)

proof: L t^(n+1)*f(t)=(-1)^(n+1)*(d^(n+1)/ds^(n+1))*F(s)

Data Structure: 1. Write a program for f(n) = 1^2+2^3+…+n^2. (i^2 = i*i) 2. If you...

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