For each of the following sequences find a functionansuch that the sequence is a1, a2, a3,...

For each of the following sequences find a functionansuch that the sequence is a1, a2, a3, . . ..

You're looking for a closed form - in particular, your answer may NOT be a recurrence (it may not involveany otherai). Also, while in general it is acceptable to use a "by cases"/piecewise definition, for this task you must instead present a SINGLE function that works for all cases.(Hint: you may find it helpful to first look at the sequence of differences of consecutive terms.)

1) 7,1/11, 15,1/19, 23,1/27, 31,. . .

2) 1, 2, 2, 3, 3, 3, 4, 4, 4, 4,. . .where each i∈N is repeated i times. (Hint: one or more of the following may be useful: floor function, Gauss sum, quadratic formula)

Expert Solution

Colby Messinger answered 2 years ago

covert the schema into 3NF. TableC (a1,a2,a3,a4,a5) functionally dependencies: a1 --> {a2,a3,a5} a4 --> {a1,a2,a3,a5} a3...

covert the schema into 3NF. TableC (a1,a2,a3,a4,a5) functionally dependencies: a1 --> {a2,a3,a5} a4 --> {a1,a2,a3,a5} a3 -->{a5} Answer: Relation1: Relation2:

Alternating Series Test. Let (an) be a sequence satisfying (i) a1 ≥ a2 ≥ a3 ≥...

Alternating Series Test. Let (an) be a sequence satisfying (i) a1 ≥ a2 ≥ a3 ≥ · · · ≥ an ≥ an+1 ≥ · · · and (ii) (an) → 0. Show that then the alternating series X∞ n=1 (−1)n+1an converges using the following two different approaches. (a) Show that the sequence (sn) of partial sums, sn = a1 − a2 + a3 − · · · ± an is a Cauchy sequence Alternating Series Test. Let (an) be...

Let A = {a1, a2, a3, . . . , an} be a nonempty set of...

Let A = {a1, a2, a3, . . . , an} be a nonempty set of n distinct natural numbers. Prove that there exists a nonempty subset of A for which the sum of its elements is divisible by n.

Question in graph theory: 1. Let (a1,a2,a3,...an) be a sequence of integers. Given that the sum...

Question in graph theory: 1. Let (a1,a2,a3,...an) be a sequence of integers. Given that the sum of all integers = 2(n-1) Write an algorithm that, starting with a sequence (a1,a2,a3,...an) of positive integers, either constructs a tree with this degree sequence or concludes that none is possible.

We say that an infinite sequence a0,a1,a2,a3,… of real numbers has the limit L if for...

We say that an infinite sequence a0,a1,a2,a3,… of real numbers has the limit L if for every strictly positive number ε, there is a natural number n such that all the elements an,an+1,an+2,… are within distance ε of the value L. In this case, we write lim a = L. Express the condition that lim a = L as a formula of predicate logic. Your formula may use typical mathematical functions like + and absolute value and mathematical relations like...

Find an arithmetic code for a symbol p4p4p3 for the source alphabet {a1, a2, a3, a4,...

Find an arithmetic code for a symbol p4p4p3 for the source alphabet {a1, a2, a3, a4, a5}, with symbol probabilities p1 = 0.11, p2 = 0.05, p3 = 0.10, p4 = 0.70, and p5 = 0.04. Show the work please.

For the arithmetic progression a1, a2, a3 .......... if a4/a7 = 2/3. Find a6/a8 ?

Consider the following eight examples: A1 = (4,20), A2 = (4,10), A3 = (16,8), A4 =...

Consider the following eight examples: A1 = (4,20), A2 = (4,10), A3 = (16,8), A4 = (10,16), A5 = (14,10), A6 = (12,8), A7 = (2,4), A8 = (8,18) The distance function is Euclidian distance. Use single-link, complete-link agglomerative clustering, and centroid techniques to cluster these examples. Show your calculations and draw the dendrograms for each technique.

2. Write the hexadecimal numbers in the registers of $a0, $a1, $a2, $a3 after the following...

2. Write the hexadecimal numbers in the registers of $a0, $a1, $a2, $a3 after the following codes running: ori $a0, $0, 11 ori $a1, $0, 19 addi $a1, $a1, -7 slt $t2, $a1, $a0 beq $t2, $0, label addi $a2, $a1, 0 sub $a3, $a1,$a0 j end_1 label: ori $a2, $a0, 0 add $a3, $a1, $a0 end_1: xor $t2, $a1, $a0 *Values in $a0, $a1, $a2, $a3 after the above instructions are executed.

Let a1 ≥ a2, . . . , an be a sequence of positive integers whose...

Let a1 ≥ a2, . . . , an be a sequence of positive integers whose sum is 2n − 2. Prove that there exists a tree T on n vertices whose vertices have degrees a1, a2, . . . , an. Sketch of solution: Prove that there exist i and j such that ai = 1 and aj ≥ 2. Remove ai, subtract 1 from aj and induct on n.

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