1. one has 2. 3. 4. Let Dn be the number of derangements of [n]. Prove that for n ≥ 1 Dn=nDn−1 + (−1)”. The generating function of the sequence (an)n is f(x)= = Use it to find a formula for an X (1-x)2(13x) Use the recurrence relation from Problem 1 to find the exponential generating function for the sequence (Dn)n. Recall that Do = 1. The following table gives the values of pk (n), the number of integer partitions of n into exactly k parts. Complete the 8th row and use it to find p(8), the number of integer partitions of 8. k 1 3 4 5 6 7 8 n 1 1 0 0 0 0 0 0 2 1 1 0 0 0 0 0 3 1 1 1 0 0 0 0 0 4 1 2 1 0 0 0 0 5 1 2 2 1 1 0 6 1 3 3 2 1 1 0 0 7 1 3 4 3 2 1 1 0 8 5. Consider the sequence (hn)n defined recursively as follows 6. ho = 1 h₁ = 2 hn=3hn-1+4hn-2 n ≥ 2. Find the generating function of this sequence. Consider the sequence (An)n defined recursively as follows Ao = [An+1 ¦ 1 = n Ai Σ n ≥ 0. (n - i)!' i=0 Find the generating function of this sequence.

6. Determine the equation for the following exponential function in the form of f(x)=ab*+ c. Show all your steps. (0, 3) -2 1- (2,0) -3 -2 -1 3 -1- -3- 2.

The equation for an exponential function is w(x) = 7 ⋅ (1.25) + 6. Which statement correctly describes the end behavior of the function? As x → ∞, w(x) → 6, and as x → ∞, W(x) → ∞ As x→→∞, w(x) → 7, and as x → ∞, W(x) - →∞. As x→ ∞, w(x) → ∞, and as X → ∞, w(x) → 7. As x→ ∞, w(x) → ∞, and as x → ∞, w(x) →→ 6.

On June 25, 2020, North Carolina reported 1009 new CoVID-19 cases.On July 14th, 2020, North Carolina reported 1956 new cases. If weassume that the growth is exponential, find the function that modelsthe number of new cases as a function of t, the number of days sinceJune 25. (Note, July 14th is t=19.) If this growth continues, how manynew cases would there be in 60 days?

Find the exponential function of the form fx) =ab* for which -3) = 7and f(O) =2.

How would I find if the sequences an=nπ cos(nπ) and an= (n2)(1/n) are convergent/divergent?