Suppose that {an} and {bn} are Cauchy sequences of R. Show that each of the following sequences is also Cauchy. (a) {an + bn}. (b) {anbn} where a,b, denotes the real number multiplication an · bn.

Q: The simple formula for a, if the first 5 terms of the sequence {an} are 1, 21, 121, 621, 3121 is:…

A: We are asked to find which is correct for the sequence.

Q: bn Уп? + 6п — п,

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Q: II(2) b, = II(2i + 1). an = e i=1 i=1

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Q: The limit of the sequence a.- (1+ (a) - (3) e) (c)e None(e)

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A: Step 1: Consider the provided question, Given, the finite sequence 2, 2, 2, 2, 2, 2.

Q: O write the expansion of : (3a+ 2b)$ , (4z – w) , - -→*

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Q: Let U be the set of first 10 even positive integers, A is set of multiples of 3 in U and B is set of…

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Q: The number sequence a1, a2, a3, .... is defined by a1 = 8 a2 = 24 and recursion formula 4an + 1…

A: We need to show that an=5n-12n-2 for a1=8 and a2=24  The given recursion formula is…

Q: Find an example of two unbounded sequences , such that their product gives a bounded sequence.

A: Consider the two unbounded sequences

Q: Let {an} and {bn} be two Cauchy sequences of real numbers. Then show that the sequence {|an − bn|}…

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Q: Let (a_n) be the sequence defined by the recursive relation a_n=a_(n−1)+5n+1, with initial term…

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Q: Rewrite 1/2(x-5) +1/4(x-5)2 +1/6(x-5)3  +1/8(x-5)4 using summation notation.

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Q: Write out the first three terms in the expansion of (x + y)^20

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Q: For 7-10 Given a_1= 3, a_2= 7 and a_n= [2a] _(n-1) + a_(n-2) for n 23, 7) Find a_3 =,* Your answer…

A: First Question: (7) Given a1 = 3 , a2 = 7 and  an = 2an-1+an-2 then a3 = 2a2+a1 = 2*7+3 = 17 Hence…

Q: Prove that product notation d/n = nτ(n)/2.

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Q: Suppose that the sequence x0, x1, x2... is defined by xo=4, x₁ = 6, and xk+2 = 4xk for k20. Find a…

A: Given Data: Let us consider the given data, xk+2 =4xk which can be written as, xk+2 -4xk=0with x0=4…

Q: 1. Find the limit of the sequence Зп+(-1)" An бn+5

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Q: Show that the converse of Theorem Limit of a Sequence is not true.

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Q: The first three terms in the expansion of (1 + ay)^n are 1, 12y, and 68y^2. Find a and n.

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Q: of all positive integers Find the sum less than 200

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Q: Suppose An = 2n^2 + 6n + 6. Find the closed formula for the sequence of differences by computing An…

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Q: Find the closed form of the generating function for the sequence 2,4, 8, 16, 32, ...

A: Generating Function: Let a0, a1, a2, a3, a4, .....be a sequence of real numbers denoted as an Then a…