Suppose f:N → N satisfies the recurrence f(n+1) = f(n) 7. Note that this is not...
Suppose that the function f satisfies the recurrence rela- tion f (n)2f(Vn)+1 whenever n is a perfect square greater than 1 and f (2) 1. a) Find f(16). . b) Give a big-O estimate for f(n). [Hint: Make the sub- stitution m log n
Please do exercise 129: Exercise 128: Define r:N + N by r(n) = next(next(n)). Let f:N → N be the unique function that satisfies f(0) = 2 and f(next(n)) =r(f(n)) for all n E N. 102 1. Prove that f(3) = 8. 2. Prove that 2 <f(n) for all n E N. Exercise 129: Define r and f as in Exercise 128. Assume that x + y. Define r' = {(x,y),(y,x)}. Let g:N + {x,y} be the unique function that...
Problem 3 (10 points) Suppose a sequence satisfies the below given recurrence relation and initial conditions. Find an explicit formula for the sequence a -6a--9a,-2 for all integers k2 2 ao = 1, a1 = 3
8. Consider the following simultaneous homogeneous recurrence relations: 3a-12bn-1 bn-an-1 + 2bn-1 for n > 1, with initial conditions ao 1 and bo - 0 (a) Find the generating function for an and then solve for an b) What is the homogeneous recurrence relation that an satisfies? (c) Repeat (a) and (b) for bn 72. 8. Consider the following simultaneous homogeneous recurrence relations: 3a-12bn-1 bn-an-1 + 2bn-1 for n > 1, with initial conditions ao 1 and bo - 0...
(1 point) (3 Marks] Note that 0 € N. Give a recursive function f:N → N that represents the sequence ao, 21, 22, az... if an 10n + 1. (1 point) [3 Marks] Find mod(31004 + 1004!, 11). A. I am finished this question
(1 point) [3 Marks] Note that 0 € N. Give a recursive function f:N + N that represents the sequence ao, 21, 22, az... if an = 10n + 1.
4. Suppose T (n) satisfies the recurrence equations T(n) = 2 * T( n/2 ) + 6 * n, n 2 We want to prove that T (n)-o(n * log(n)) T(1) = 3 (log (n) is log2 (n) here and throughout ). a. compute values in this table for T (n) and n*log (n) (see problem #7) T(n) | C | n * log(n) 2 4 6 b. based on the values in (a) find suitable "order constants" C and...
5. Let F(n, m) denote the number of paths from top-left cell to bottom-right cell in a (n x m) grid (that only permits moving right or moving down). It satisfies the recurrence relation F(n, m) F(n-1, m) + F(n, m-1) What should be the initial condition for this recurrence relation? (Hint: What would be the number of paths if there was only a single row or a single column in the grid?)[5] Convince yourself that F(n, m) gives correct...
Let f:N + N be defined by the recursive definition: Base case: f(0) = 7 Recursive step: 3nf(n-1) and g:N + N be defined by the recursive definition: Base case: g(0) = 1 Recursive setep: g(n) = 8 * g(n-1) +4n Find a closed-form definition for fog(n)
4. Define a function f:N → Z by tof n/2 if n is even 1-(n + 1)/2 if n is odd. f(n) = Show that f is a bijection. 11 ] 7. Let X = R XR and let R be a relation on X defined as follows ((x,y),(w,z)) ER 4 IC ER\ {0} (w = cx and z = cy.) Is R reflexive? Symmetric? Transitive? An equivalence relation? Explain each of your answers. Describe the equivalence classes [(0,0)]R and...