Let an be the sequence defined recursively as follows: a1 = 1 a2 = 1 Prove that an E [1, 2] for all n. an = an-1 2 + 1 an-2 for n ≥ 3.

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A: The solution is given as

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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.

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Q: 5. Suppose b₁,b2, b3, ... is a sequence defined as follows: b₁ = 4, b₂ = 12 bk = bk-1 + bk-2 for all…

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Q: Give a recursive definition of the following sequences {an} ,n=1,2,3,..... 1. an =2n+3 2.  an =n+2…

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Q: Use mathematical induction to prove that for each integer n2 0, if F, F F ... is the Fibonacci…

A: (i) Consider the Fibonacci sequence F0, F1, F2, … where Fn=Fn-1+Fn-2 and n≥0. We have F0=1, F1=1. We…

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Q: Define the following sequence recursively, giving two instructions in mathematical language. {3, 6,…

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Q: 3. Find the unique sequence (an) of real numbers with Σk-0 akan-k = 1 for all n ≥ 0.

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A: Let b0, b1, b2,...... be the sequence defined by  bn=C(3)n+D(-2)n for every integer n≥0,  where C…

Q: = 3 · gn-1 + 2n, for n > 1. If the theorem %3D The sequence {9n }is defined recursively as follows:…

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Q: 5. Find the first four terms of the following recursively defined sequence: bk = bk-1 + 3k, for all…

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