Q3. A sequence of integers o, ₁, is defined by xo = 2,₁ = 5 and m = xm-1 + 2xm-2 for m≥ 2. Prove that for all n E Z with n ≥ 0, Xn = 7.2 + (-1)+1 3

Q: Find the length s of the path (3 + 10r,9 + 2r312) over the interval 5 < t < 6.

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Q: 5. Suppose N > 1. (i). Prove that N is of the form 4n +1 if every prime factor of N is of the form…

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

Q: n3 + 3 n5 n4 - Prove that 14 + 24 + ..n* = +. n for all n > 1. 30 -

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Q: If all tems except ys of the sequence Y1, Y2» Y3•…· Y9 be given , show that the value of ys is 56…

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Q: 8. The (2*+1 – 2k) = %3D k=0

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Q: A sequence is said to be decreasing sequence if Vn >1 2n.a O n+1 'n²³n+1·bO n+ n+1dO n+ 1

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Q: 5. Prove that, for n> 1, n

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Q: 5.3.2 (a) Suppose that n > 3. Prove that ("+1)ʻ 5.

A: Given:-             n+1n2 &lt; 2 n+1n2 -22&lt;0n+1n+2n+1n-2&lt;0n+1n+2 =…

Q: 5. (*) If a and b are nonzero integers such that a | b and b|a, prove that a = ±b.

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Q: 1. +1 Select one: a. th n! b. tn-1 (n 1)! C. n! tn+1 d. n! th-1 e. t+1 (n +1)!

A: Solution:-    L-11sn+1  ∵we know  that:-By using Laplace properties:-

Q: (d) p+3 E (3k2 – 2k + 1). k=-2 (e) р-1 E (3k? – 2k + 1). k=-2

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Q: What is the summation from n = 1 to infinity, of  (3n+1) / (5n + 4) ?

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Q: * 30. Let n and k be integers with 1 ≤ k ≤n. Show that n Σ() ( k=1 k n - 1) = (2n+2)/2 - (2n).

A: The answer for the given questions has been solve step by step below please find the attachments…

Q: 3 4 N + 1 5) Show that the sequence 7 2N + 1 is bounded and monotonic.

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Q: a (a) For all ne N and a # 0, show that an k=1 (b) For a sequence (xn) in R, suppose that there…

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Q: (a) Let n EN. Determine the set of all integers which leave a remainder n+1, n+2, n+3 when divided…

A: PART A As it is given that 2n+1, 2n+3, 2n+5 are consecutive odd integers so they are pairwise…

Q: C. Prove the following using PMI. 1. 1+4+7+... + (3n-2) = " (3H=1 for all n21. n (n+1) 2. 1+2+3+...…

A: We will prove the given statement using PMI.

Q: (1.2) Show that m m+n+1] m k=0 for any two positive integers m and n.

A: Please check the next step for detailed answer.

Q: Use |S – SN| < bN+1 to find the smallest value of N such that SN approximates the value of the sum S…

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Q: all n N we have We apply this successively for n = bn+1 bn N, N+1, n 9 bN+1 < (m + €)bN bN+2 < (m+…

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Q: 6. For any positive integer n > 1 one has: (2n)! (n!)² n+1 <

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Q: 2 the ratio of the sum of m and n terms of an AP is m*:n“ ,show th " is (2m – 1):(2n – 1) th that…

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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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Q: 5 Fn+1 3 F,. Compute R6 Define a Sequence R,

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Q: 5. For n ≥ 1, establish the formula (2n + 1)² = (4tn + 1)² – (4tn)².

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Q: Let a, := 7+ Vn , n 2 1, be a sequence of real numbers. Find an n(K) EN Ə a, > K whenever n> n(K).…

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Q: 6. Let {an}1 be a sequence. If a1 = 1 and An+1 :2an + 1, use induction to prove an = 2" – 1. %3|

A: given that:-              an =1 and an+1 =2an+1  now, to prove  an=2n-1 we use induction.

Q: For items (14)-(15), find a number that provides a counterexample to show that the given statement…

A: 14.  We need to find a number that provides a counter-example to show the given following statement…

Q: -3an The solution of the recursion relation ant1= n > 0, with ao = 1, (n+1)? " is an O None of these…

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Q: In the inductive step, we assume that for any k > 10 , P(j) is true for any 8 <js (k-2) and we will…

A: The proof by mathematical induction consists of two steps: The proof consists of two steps: The…

Q: Let n E N* and a, BE R. 1 a. The sequence B (T-4)"-a is neither increasing nor decreasing b. The…

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