Question

Use 3 and 4 as the first two terms of a Fibonacci-type sequence.
(a) Find the next five terms?
3, 4, ___7____, ___11____, ____18___, ___29____, ____47___
(b) Write a symbolic description for this Fibonacci-type sequence using variables tn-1 and tn+2 as terms of the sequence.

Answer 1

are 7, 11, 18, 29, 47 correct

Answer 2

Those are the correct for part a
for part b though, did you mean
\[t _{n-1} \] & \[t _{n-2}\]?

Answer 3

That is what the teacher wrote I have not answered that part yet, it that incorrect?

Answer 4

if the teacher did want t_n in terms of t_(n-1) & t_(n+2), then the problem becomes a bit tricky. The easiest way to solve analytically is to use the fibonacci sequence definition of\[t _{n}=t _{n-1}+t _{n-2}\] for n+2 and n+1:\[t_{n+2}=t_{n+1}+t_n\]\[t_{n+1}=t_n+t_{n-1}\] and solve both of those in terms of t_(n+1)\[t_{n+1}=t_{n+2}-t_n\]\[t_{n+1}=t_n+t_{n-1}\]. Setting those equal to each other and solving for t_n we get\[t_{n+2}-t_n=t_n+t_{n-1}\]\[t_{n+2}-t_{n-1}=2t_n\]\[t_n=(t_{n+2}-t_{n-1})/2\]

Answer 5

im confused

Answer 6

which part are you confused about?

Answer 7

I confused myself because I was looking at both formulas thinking it was solved instead of realzing they are two seperate terms

Answer 8

Do you understand it now, or do i need to clarify a step or two?

Answer 9

How did you get the n+2 and n+1?

Answer 10

n+1 and n+2 are representing the next two terms, respectively. as far as how i derived the two formulae when i defined t_{n+1} and t_(n+2}, by taking the original Fibonacci function: \[t_n=t_{n-1}+t_{n-2}\] and shifting it over two.
step 1: replace n with another letter, I will use p.
\[t_p=t_{p-1}+t_{p-2}\]
step 2: shift two to the + by setting n+2=p
\[t_{n+2}=t_{n+2-1}+t_{n+2-2}=t_{n+1}+t_n\]
you can use the same process for shifting it 1 to get the second equation i listed.
hope that helps (:

Answer 11

Yes this help a lot thank you