Take any of the generalizations about the Fibonacci Numbers that we have considered, and investigate what happens if the sequence is formed by two different starting numbers but continues in the same way by adding successive pairs of terms. For example, you might form a new pseudo-Fibonacci sequence in this way:
2, 5, 7, 12, 19, 31, 50, 81, 131,.....

a) Does the generalization still hold? Why or why not?
b) Can you generalize for all Fibonacci-type sequences?
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I have 3 generalizations:
1) f_n = 2f_(n-1) - f_(n-3)
2) f_(n+4) + f_(n+1) = 2f_(n+3)
3) summation from of f_i where i from 1 to n = f_(n+2) - 1

Note that: "f_n" means f sub n

I tried them out. The 1st and 2nd generalization still holds, but the 3rd one doesn't hold. So, my answer for question a) is "yes" for 1) and 2) but "no" for my 3 rd generalization. My instructor said that I am wrong. And she said that if I look, for example, at # 3 more carefully there is a generalization but I have to "tweak" it a bit but that is because the numbers I start with are different. I don't see what she meant by tweaking it. Anyway, can you help me with this problem in details please?

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The solution is attached below in two files. the files are identical in content, only differ in format. The first is in MS Word XP Format, while the other is in Adobe pdf format. Therefore you can choose the format that is ...

... The proof is complete. Step-by-step computations and explanations are shown for both problems. The expert provides proofs of the Fibonacci Sequence. ...

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... C1 = , C2 = ( 8) 25 25 Thus, the explicit form of Fibonacci sequence will be ... 1 ( 12) ϕ 5 Now, the expression required by the problem will be ...

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(See attached file for full problem description ... respectively b) prove that f12+f22+..fn2 = fnfn+1 whenever n is a positive integer fn is the Fibonacci sequence. ...