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- Find a recurrence relation with initial condition that uniquely determine the sequence 6, -18, 54, -162, ... 20147. Find the first five terms of the sequence defined by each of these recurrence relations and initial conditions. a) an = 6an-1, a0 = 2 b) an = a2 n-1, a1 = 2 c) an = an-1+ 3an-2, a0 = 1, a1 = 2For the sequence 2, 9, 16, 23, 30, 37, 44, 51... find (a) closed formula (b) recurrence relation and initial term.
- The sequence {a} begins with 2, 1,-1, -2, -1, 1, 2, ... n From the list below select a recurrence relation and initial conditions that define the given sequence. {a}: 2, 1, 1, -2, -1, 1, 2, ... n O a = 2, a₁ = 1, a = a₁ 1 n n-1 O a = 3, a₁ = -1, a = a 1 n a=2, a₁ =1, a n a n-2 a=2, a₁ = 1, a = a n -an-2 + a n-2 .a n-1 n-1 n-2-an-1From the set of all 10 digit sequences, that consist only the numbers 0, 1, 3, 4, randomly choose one sequence. What is the probability, that the sequence a) consists exactly three 1's, b) consists exactly six O's, two of them in the ends of the sequence.Write a recurrence relation that describes the sequence 1, 1,3, 3,5,5, 7,7,9,9, .... Don't forget to specify any initial conditions.
- 12. Show that the sequence {a} is a solution of the recurrence relation an = -3an-1 + 4an-2 if a) an = 0. c) an = (-4)". b) an = 1, d) an = 2(-4)" +3.Suppose I’m placing 6 volumes of a mathematics book series on my shelf, but I’m in ahurry and my office is a mess, so I just put them on the shelf randomly. Assume that allpossible orderings of the 6 volumes are equally likely to occur. Also, note that each volumeis numbered “Volume 1”, “Volume 2”, etc.Let X = the number of volumes that are in the correct spots before the first incorrectlyplaced volume. For example: X = 3 for the arrangement 1, 2, 3, 5, 4, 6, since the first3 volumes are placed correctly, then the 4th spot has Volume 5 in it. Another example:X = 0 for 2, 6, 3, 4, 5, 1, since the very first spot has the wrong volume in it!(a) Find the support for X.(b) Compute the probabilities that X = x for each value x in the support in X.(c) Use the probabilities to determine the formula for the PMF as a singlefunction pX (x) = P (X = x) with input xFind a recurrence relation for the following sequence: 1, 2, 6, 24, 120, 720, ...