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6. Find a recursive definition for the following sequences defined by the closed formulas: (a) an...
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)
2. [5 Pts] Give a recursive definition of the sequences a1, 12, ... described by the following explicit formulas: (i) an = 6n-5, (ii) an=2" - 1.
4. Find the first six terms of the sequences defined by each of these recursive definitions: (a) ao = 1; a1 = 1; an = An-1 + 2an-2 for n>2 (b) bo = 1; bı = 2; bn = 2bn-1-bn-2 for n> 2 (c) co =1, cı=2; Cn Cn-2-0-1 for n> 2
4. In lectures, we defined closed subsets of Rn. The definition can be generalized in the following way. Let X be a subset of R". We say that a subset S C X is closed in X if all limit points of S that are in X are also in S. [Any closed subset of Rn is "closed in Rn*) State whether each of the following sets S is closed in X. For cases where X - Rn (including the...
(14) Given a sequence of integers {fi,f2 /. defined by the following recursive function f (n)-., n e N such that s(2) 5, Evamine the sruture of this sequence and then compute J, in closed-form.Prove by using strong induction as discussed in class that your function f, is indeed correct.
discrete math
Write a recursive definition that generates the terms of each of the two following integer sequences: 9. a. 1,-1,2,-2, 4,-4,8,-8, 16,-16, b 1,2, 3,6, 11, 20, 37, 68, 125, 230, 423,. Hint: A recursive definition, which comprises initial conditions and a recurrence relation, is discussed in Section 5.6 of our textbook ] 10. Use a truth table to determine whether the following argument form is valid. Include a sentence or two referring to your truth table to support...
11) Find a closed form for the generating function for each of the following sequences: a) ?? = 5(−2)? g) 0, 0, 0, 4, -12, 36, -108 h) 0, 1, 0, 4, 0, 16, 0, 64, 0, ...
(6 points) Use repeated substitution to find an exact (non-asymptotic) closed-form ex- pression (a non-recursive function of n) for the recurrence F(n) - 12 F(n-1)+2 n2 2 Show your work, and write your final answer in the box at the bottom of the page.
Discrete Mathematics
Given the following recursive definition of a sequence an do = 2 a = 9 an = 9an-1 - 20an-2, n 2 2 Prove by strong induction that a, = 4" + 5” for all n 20.
) Find a closed form for the generating function for each of the following sequences: an=5-2n 0, 0, 0, 4, -12, 36, -108 0, 1, 0, 4, 0, 16, 0, 64, 0, …