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, . .., con- structed by including the integer k exactly k times. Show that an = V2n +

Q: Evaluate the limit of the sequence or state that it does not exist. an = (2n + 5n + 1)/(5n)

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Q: Let {rn} and {Yn} be two sequences defined by In+1 = 2 2 xn - 1, and Yn+1 = 2xn + 6, for each…

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Q: 11. Let (an) be the sequence defined recursively by, a1 = 3, a2 = 4, for n > 3, an 2ап-1 + аn-2.…

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Q: let a,=!. Consider the sequence dany generaed as follows: = It I Itan a ntl (a) Write down expresian…

A: We will answer the first two questions since we don't answer multiple questions at a time. Please…

Q: Find a function f(n) that identifies the nth term anan of the following recursively defined…

A: Find a function f(n) . Given that a1 = 1 and an+1= (n+1)an for n>=1. Here i attached solution of…

Q: Evaluate the limit of the sequence or state that it does not exist. an = (8n)/(n!)

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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: 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. Let D ⊆ R and (Xn) be a sequence in R. Show that Xn → x such that x ∈ D if and only if D is…

A: The problem is solved with the concept of limit point of a set in R. Every part is elaborated in the…

Q: Let (an), (bn) be sequences, so that (b,) is defined as b, = an – an+1· Prove that the series b, n=1…

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Q: Prove {3n/(3n-1)} n=1 to infinity, is a decreasing sequence in R.

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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: 6. * Let {a,}nEN be the sequence defined by 6(a+1) a + 11 aj = 4, ant+1 = for any n >1 Given that…

A: We know that every monotonic and bounded sequence is convergent

Q: 4. Define the sequence an by a, = 1, a, = 33, and an = 3an-1+ 10an-2 for n 2 3. Prove that an = 5" –…

A: Concept used :- we use mathematical induction to solve the given question.

Q: Define the sequence {an} as follows: aj = 2 an = an-i + 2n for n > 2

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Q: Evaluate the limit of the sequence or state that it does not exist. an = ((1 + (3)/(n))2n)

A: Evaluate the limit of the sequence or state that it does not exist.

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,…

A: The given sequence is 1,2,2,3,3,3,4,4,4,4,..., The first integer 1 occurs in first position. The…

Q: B. Express a, in terms of n for each of the following sequences. 1. 7, 8, 9, 10,.. 2. 3, 7, 11, 15,.…

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Q: Find the limit of the sequence {an} if 1 - (1)/(n) < an < 1 + (1)/(n), forevery integer n ≥ 1.

A: Given: 1-1n<an<1+1n where n≥1 We have to find the limit of the sequence an:

Q: 8. Consider the sequence {an} such that an+1 = -3 – 2an. Then lim an = -1.

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Q: Give a recursive definition of the sequence {an}, forn= 1,2, 3, . where an = n². %3D .... O a. a1 =…

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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: Find the limit of the sequence or determine that the limit does not exist. n! an= 4n.7n

A: Consider the given sequence, an=n!4n·7n

Q: If (xn) is a Cuachy sequence in R, then it is unbounded. Select one: OTrue OFalse Every sequence of…

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Q: Define a sequence (an) recursively by setting a1 = V2 and an+1 = v2+ an for n > 1. (a) Show that an…

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Q: 13. Let (an)=I be a Cauchy sequence, and let p : N → N be a one-to-one function. Show that the…

A: A sequence is called a Cauchy sequence if as the sequence progresses, the terms of the sequence…

Q: 6. Given a sequence ro, z1,2,... which satisfies the relation z,= 20 = 0, a1 = 2. Let 4zn-1- 3zm-2…

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Q: Let a sequence be defined by a, =1,a, =1. and a =a+a2 for all n > 2.

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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: if a, is a sequence and lim a, = ∞ then the sequence diverges Select one: O True O False

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Q: Evaluate the limit of the sequence or state that it does not exist. an = (n2 + 4)/√(4n4 + 1)

A: We have to find the limit of the sequence: an=n2+44n4+1 So finding limit, limn→∞n2+44n4+1 As it is…

Q: Suppose b1, b2, b3, ... is a sequence defined as follows:  b1 = 4, b2 = 12,  bk = bk-2 + bk-1 for…

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Q: Consider the sequence recursively defined by bị = 3, b2 4, and bn+1 2b, + bn-1 for n > 2. Show that…

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Q: The generating function for the sequence 1,0.1,0.1,0,1 is 3)1 +x + x + + x+ x5 b)1 +x2 - 4. )1+ x+ x…

A: generating function is a way of encoding an infinite sequence of numbers (an) by treating them as…

Q: 1. Give a recursive definition of a sequence an, n > 1, where an 5n – 7

A: Please see the next step for solution.

Q: Give a recursive definition of the sequence {a,} , n = 1 , 2, 3, ... if an 6n %3D OA. an+1 a, for…

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Q: Prove: if a sequence is defined recursively by a₁ =1 and an = n/(n-1) an-1 for n≥ 2 then an-n for…

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Q: Consider the sequence {an} c Q given by So n = 2, 4, 7, 17 n+1 2n otherwise In the context of the…

A: an=0n=2,4,7,17n+12notherwise

Q: Calculate the lim sup and lim inf (if they exist) for the sequence (a,) given by: a, =7+(-1)".

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Q: Evaluate the limit of the sequence or state that it does not exist. an = ((2n + 2)!)/((2n)! n2)

A: Consider the given nth term of the sequence, an=2n+2!2n! ·n2 Since, 2n+2!=2n+2· 2n+1· 2n! Then,…

Q: 17 a) Determine the sequence generated by f(x) = 6e5x – 3e2x b) Use generating functions to…

A: Given that (a) f(x)=6e5x-3e2x To determine the sequence generated by f(x): (b) To determine how many…

Q: The generating function of the sequence 5,5,5,5,5 is 1-x5 O (1-x) 5 1-5x° O (1-x) 1-125x5 (1-x)…

A: We need to find the generating function of the sequence 5,5,5,5,5.

Q: Consider a sequence defined on the set of non-negative integers as follows: an = -2(-1)" for n 2 0.…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Let us define the sequence called a, so that a1=3 a2=5.   a_n = a_(n−1) + 2_(an−2) − 2. for each n…

A: We use Strong Induction to prove our claim.

Q: Find the limit of the sequence or determine that the limit does no In(n + 8) an n1/n does not exist…

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Q: Suppose that ao, a1, a2, ... is a sequence with these properties: ao = 6, a1 = 17, and an = 5an-1 –…

A: Solution is given by induction :

Q: Let ao, a1, a2, . .. be a sequence defined recursively by ak-1 + 6ak-2, ak for each integer k > 2…

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Q: See image

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