1. The Fibonacci numbers are defined by F1 = 1, F2 = 1, F3 = F1+F2 = 2, and, in general, for n > 3, Fn = Fn-1+ Fn-2. Thus, the Fibonacci sequence is 1,1,2, 3, 5, 8, 13, 21, 34, .... What is the sum %3D F1 + F3 + ... + F2n+1? and the sum F2 + F4 + ...+ F2n? Prove your answers by induction on n. 2. Show that (i) 2| Fn 3| n, (ii) 3| Fn 4|n, (iii) 4 | Fn + 6 | n. Hint: For example in case (ii), write Fn = 3qn + rn with 0 < rn < 3 and consider how the Tn's are related.

Q: 2. The Fibonacci sequence (fn) := 0, 1, 1, 2, 3, 5, 8, . is an example of a recursive sequence with…

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Q: In the Fibonacci sequence {0, 1, 1, 2, 3, 5, 8, …}, what is the value of F5

A: Given that the Fibanacci sequence 0,1,1,2,3,5,8 we know that The reccurence relation of the…

Q: 2. For the sequence 1, 5, 12, 22, 35, 51, ... i) Use subscript notation to write a recursive formula…

A: The explicit rule for a sequence is used to define the general term of a sequence. For example, the…

Q: What is the sum of the sequence E(1+k!) R-0

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Q: The Fibonacci numbers are a sequence numbers defined by F₁ = 1, F₂ = 1, and Fn = Fn-1 + Fn-2 for n ≥…

A: Since, the Fibonacci number is defined as F1=1,F2=1 and…

Q: The sequence a,,a,a...... is defined as a₁ =n² +3n nez' i. Find the first four terms of the…

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Q: Suppose that g1, 92,93, ... is a sequence defined as follows: 91 = 3, 92 = 5, gk = 3gk-1 - 2gk-2 for…

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Q: Calculate an for the sequence 2/(12+1) +1/(22+1) +2/(32+1) +1/(42+1) +.

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Q: 2.) Write the general formula for un for the arithmetic sequence 12, 4, –4, –12, ...

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Q: 12. Find the first 5 terms of the sequence: 2n a, =(-1)" n +1 a) b) a, = 4a-5 and aj =1 | n-1

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Q: 2) Consider the sequence of natural numbers: 1, 1, 2, 3, 5, 8, 13, 21, .... This is the famous…

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Q: Suppose the sequence goes; 4, 8/3, 16/9, 32/27,... please find the sum of the first 25 even terms

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Q: (b) Define another recursive sequence (hn) by removing fi, f3, f5,... from the bonacci sequence.…

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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.