**) Define a sequence a1, a2, a3, ... as follows: a1 = 1, a2 = 3, and a, = an-1+ an-2 for al ategers n > 3. (This sequence is known as the Lucas sequence.) Use strong mathematica duction to prove that a, < ()“ for all positive integers n. %3D

Q: 1-Let B be the set of all infinite sequences over {0, 1}. Is this set countable or not, prove it?

A: Please post the multiple parts separately.Here I answered the 1st question as per our policy.

Q: Determine if the sequence is bounded and if it is monotonic. Determine if the sequenc converges and…

A: The given sequence is: an=(-1)nn100 Now, the terms of the sequence are: a1=-1,a2≈1.00,…

Q: Prove {91 + 1/n} n=1 to infinity is a Cauchy sequence using the definition of a Cauchy sequence.

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Q: Let Fn be Fibonacci sequence. Prove that:

A: By the definition of the Fibonacci sequence, Fn+2=Fn+1+Fn and the Fibonacci sequence is satisfied…

Q: Determine whether the sequence converges In n 2) a, n

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Q: Can you find a formula or rule for the nth term of a se- quence related to the prime numbers or…

A: The formula for nth term of sequence for given problems are given in step 2.

Q: Let a, be the nth term of the sequence 1,2,2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6,6,…

A: Let, the sequence be an Here, a0=1,a1=a2=2,a3=a4=a5=3, and so on.

Q: Prove that y is a negative Cauchy sequence if and only if -y is said to be a positive Cauchy…

A: We need to prove that y is a negative Cauchy sequence if and only if -y is a positive Cauchy…

Q: 3.) Determine whether the sequence is monotonic and whether it is bounded. sin /n

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Q: determine whether the sequence is monotonicand whether it is bounded. an = (3n + 1)/(n + 1)

A: To check the given sequence is monotonic: Consider Take derivative; We see that the first…

Q: 2. Using the definition of a Cauchy sequence, prove that (1/n2) is a Cauchy sequence.

A: We have to prove that given sequence is a  Cauchy- sequence , so by definition of Cauchy sequence-

Q: 6. Determine if the sequence is geometnc. DRHIS,find the comman fatio. an-1,3,-1.. ماتوقو

A: Given: 27,-9,3,-1,..............

Q: Prove that the sequence {n³ + 1/n³} is a cauchy sequence.

A: The given sequence is an=n3+1n3. A sequence an is said to be Cauchy sequence if for each ε&gt;0,…

Q: a) Show that the sequence a, = (-1)" +- does not converge. b) Let b, be a sequence such that for all…

A: Given the sequence an = ( -1 )n + 1/n . It has two subsequences a2n = 1 + 1/n converging to 1 and…

Q: Can you find a formula or rule for the nth term of a quence related to the prime numbers or prime…

A: Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts…

Q: 2. Give an example of an unbounded sequence {an} of negative terms such that an does not diverge to…

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Q: determine whether the sequence is monotonicand whether it is bounded. an =(2n + 3)!/(n + 1)!

A: To find the sequence is monotonic or bounded. Consider

Q: how whether the following infinite sequences converge or diverge : 00 {an} = 2) {bn} = {Vn6 + 12n³ –…

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Q: Prove that there is no hooked Skolem sequence of order n= 8. Type your answer here

A: We need to prove there is no hooked Skolem sequence of order n = 8. We know the theorem, A hooked…

Q: Determine whether the sequence with the given nth term is monotonic and whether it is bounded. Use a…

A: Given that: The sequence an = 3n( n + 2) .

Q: Define a sequence by recursion An + an+1 an+2 = with initial conditions aj = ì, a2 = µ where 1, u E…

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Q: Which sequence could be described by the first term: Aj = 1 and the recursive (an)² + 1 definition:…

A: Given, an+1=an2+1 a1=1

Q: Which of the given infinite sequences is both an increasing and a finite sequence for every n≥1?

A: a sequence is increasing if an+1 ≥ an for all n ≥1   We have to check one by one which one…

Q: Suppose that a sequence is defined a=-3, + 10 n-1 List the first four tor mc of th

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Q: State whether the sequence, whose nth terms are indicated, is bounded and whether it is eventually…

A: let  an=n9n

Q: Consider the sequence with general term n! An nn Show that the sequence is bounded and monotonic…

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Q: Use the definition of a convergent sequence to show {7-4/n} n=1 to infinity, converges to 7 where…

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Q: 3. Define Cauchy sequence and prove that the sequence { } is a Cauchy sequence in C[1,3]. nt+5

A: According to the given information, it is required to define Cauchy sequence and prove that the…

Q: (a) Define a sequence by ai = 7, and ak+1 = Vak for all k > 1. Does ak converge or diverge? Explain.…

A: We will be using some simple and well known results of series to arrive at the final solutions

Q: (a) Let {n} be a sequence such that there exist A> 0 and c € (0, 1) for which n+1- n ≤ Ac" for all…

A: As per the policy, we are solving first question. Please repost the question and specify which…

Q: Let a, be the nth term of the sequence 1,2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6,…

A: Let an be the nth term of the sequence 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6… constructed by…

Q: Find at least three different sequences beginning with the terms 1, 2, 4, whose terms are generated…

A: According to the guidelines I am answering first question only . Kindly post other questions…

Q: The nth term of the sequence in number 1 is given by a, = (1+v5)"-(1-v5)" and the ratio, , between…

A: Here, the given sequence is: an=(1+5)n-(1+5)n2n5 And also given that sequence anan-1 converges to…

Q: Determine if the sequence is monotonic and if it is bounded. n! 5n

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Q: how that the sequence (1 – (-1)" +1/n) is divergen

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Q: Let (n) be a sequence defined by 2₁ = 1 and for n ≥ 2, 1 2+²-1 In Prove that (n) is Cauchy using the…

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Q: Suppose that f0, f1, f2, . . . is a sequence defined as follows:f0 = 5, f1 = 16,fk = 7fk−1 − 10fk−2,…

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Q: The first four terms of the sequence defined by An = nan-1+n'an-2, ao = 3, aj= 5 are ao = 3, a1 = 5,…

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Q: Show that (1/n²+ n+1)!nEN is a Cauchy sequence.

A: Solution :-    The given sequence is              an =   1/( n2 + n + 1 )     ;  n belong to N .…

Q: Determine whether the sequence converges or diverges an = In(n + 1) – In(n) %3D lim an %3D n- 00

A: Given sequence is an=lnn+1−lnn We have to find the limn→∞an Use property lnm−lnn=lnmn…

Q: Prove that the following sequence is cauchy sequence by definition (xn) = Vn +1– Vn Hint :…

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Q: 6. When a, = n(-1)", determine whether the sequence {a„} is increasing, %3D decreasing, or not…

A: We have been given the sequence an=n(-1)n

Q: Let an be the nth term of the sequence 1,2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6,…

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Q: 35. Prove that the sequence {n!}^, beats the sequence {2"}-1-

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Q: 4. Define a sequence by an+1 = -an?, with -1 < ao < 0. Prove that {an} is increasing and bounded…

A: Consider the terms of the sequence put n=0 a1=-a02 here -1≤a0&lt;0 So -1≤a1&lt;0 but certainly a1≥a0…

Q: Define a recursive sequence by a1 = 2 and an+1 = 1 − a2n. Either prove that (an) converges or prove…

A: Define a recursive sequence by a1 = 12 and an+1 = 1 − an2. Check the second term: a2 = 1 −122=34 It…

Q: 5) Consider the infinite sequence defined as follows: B1 = 4 and, for each integer n> 1, B, = 3(…

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.