1. For the function defined recursively by f(0)=5 and f(n)= 4f (n-1)+3, answer the following: a. Find a closed form representation for this function. Your closed form should not include any series.
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- 4. Find a closed form representation for the function defined recursively by f(0)=5 and f(n+1)=3f(n)+4. Prove that your representation is correct using an inductive argument.Determine whether the proposed definition isa valid recursive definition of a function f from the setof nonnegative integers to the set of integers. If f is welldefined, find a formula for f(n) when n is a nonnegativeinteger and prove that your formula is valid. f(0) = 1, f(n) = −f(n − 1) for n ≥ 1Find a closed form representation for the following recursively defined function. Give the run-time complexity of each recursively defined function.
- 7. Find a closed form representation for the function defined recursively by f(1)-10 and f(n)=5ƒ(%) + n.Problem 1. Prove that the following functions are Primitive Recursive. I – 1 if x > 0, (1) mPred(x) = for x € N. if x = 0. if x > 0, (2) sgn(x) = for x E N. 0 if x = 0. if x > 0, (3) sgn(x) = for x E N. 1 if x = 0.Implement the XOR function exclusively through a combination of AND, OR, and NOTperceptrons introduced in class. (Note: Non-linear activation functions are prohibited at thispoint.) Also, provide a detailed explanation of your implementation with a summary of your implementation, theresults, and your analysis.
- 7.) Given the following Truth Table for the function F, generate a Karnaugh-Map for the function F (1 Point), and a Boolean Algebra representation of the function F (1 Point). Each variable (A,B,C,D) and its complement (Ā,B, C,D) are available as an input if needed. I provided a blank K-Map template to save you time. A BC D F 1 0 0 01 0 01 0 0 0 1 1 0 10 0 0 1 0 1 0 11 0 01 1 |1 10 0 0 10 0 1 1 0 1 1 10 10 1 |1 110 0 1 10 1 11 1 0 1 1 1 1 1 1Construct a subprogram that solves the differential equation set dy g(y,t)dt dt with the fourth-order Runge-Kutta method with different parameters from those given in the text. Use the total number of components in y and the initial condition y(0) = yo as the input to the subprogram. Test the fitness of the parameters by comparing the numerical result from the subprogram and the known exact result for the motion of Earth.Prove:Church’s thesis: The class of computable functions is equal to the class of intuitive computable functions.
- For each of the following functions, indicate the class (g(n)) the function belongs to. (Use the simplest g(n) possible in your answers.) Please show work to prove the assertions.Let ∑={a,b} and T be the set of words in ∑* that have an equal number of a’s and b’s. (a) Give a recursive definition for the set T. (b) Show that abbaba is in T by building up from the base case through recursion. (c) Is your recursive definition uniquely determined?Give a recursive definition of the relation is equal to on N x N using the operator s.