1. Let a, = 1 and an+1 = 1+ 1+an (a) Find the first six terms of the sequence {an}. (b) Prove that {an} is bounded. (c) Prove that {a1,a3,a5, ... , A2n+1» …..} is increasing and {a2, a4, a6, ... , a2n, ...} is decreasing. (d) Find lim an.

Q: For the sequence an = n +1' its first term is its second term is its third term is its fourth term…

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Q: Find two different explicit formulas for the sequence {1, -2, 3, -4, -5, ⋅ ⋅ ⋅}.

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Q: Find the generating function for the sequence 1,−2,4,−8,16,….

A: The general expansion of the binomial (1-x)-1 will be, (1-x)-1=1+x+x2+x3+….

Q: 30. а, 2 1+ %3D - n

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Q: 10. (6 points) A sequence is given by the recursive definition an = an-1 + 2, ao = 3. a) Find a₁,…

A: The objective of the question is to find the first three terms of the sequence defined by the…

Q: 25. Write the nth term of the sequence 2, 8/3, 32/9, 128/27... as a function of n.

A: Given,          2,  83, 329. 12827, . .  .

Q: -) Determine the first six terms of the sequence defined by t₁ = -5 and t = 3t-1 + 8. 5, 23, -61,…

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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: Find the limit of the sequence. {v5, y5v5, V 5V5V5...}

A: Given sequence  5, 55, 555, …. We have to find its limit.

Q: 35 (b) (5+7k) = k=D1

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Q: 1. Find the limit of the sequence {√3, √√3+√√3, √√3+ √√3 + √3,...}.

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Q: 2. Let an be the sequence defined inductively by a₁ = 2 and an+1 == 1/2 (a₁ + ²2² ). an (a) Prove by…

A: a. By induction, we can conclude that an € [1,2] for all  n € N.b. It is proven that an2 >= for…

Q: Find an increasing subsequence of maximal length and a decreasing subsequence of maximal length in…

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Q: f(n)=(n-1/2) (n-1/3) Its n-1 over 2 and n-2 over 3 Write the first six terms of the sequence…

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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: inf{a,b, : n e N} > (inf{a, : n € N}) × (inf{b, : n E N}). (1) Does the inequality (1) continue to…

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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.