Fibonacci Problem help please?? Pn=(2/3)pn-1 + (1/3)pn-2

[dripnoodle]

Hey guys I need help with solving the following problem. I have solved part 1 and 2 and got the following answers 60 and 55 respectively. Need help for part 3 and 4.


Pn=(2/3)pn-1 + (1/3)pn-2
Now suppose there were 90 plants the first year and 45 plants the second year.
1. How many plants are there in the third year?
2. How many plants are there in the fourth year?
3. Using the recursion relation Fn=F n-1 + F n-2 , we discovered that the limit ratio of the Fibonacci sequence was one half of one plus square root of five, which is approximately 1.6. This means that, eventually, each number in the sequence is approximately 1.6 times the one before. As a consequence, the numbers get arbitrarily large as we look further and further out in the sequence. Questions:
(a) If the limit ratio had been 1⁄2, what could you say about the numbers far out in the sequence?
(b) What if the limit ratio was 1? 3. Find the limit ratio for the sequence. (Same as problem 38 in the Homework with different numbers; you should mimic what was done in class to get the golden ratio – the limit ratio for the Fibonacci sequence.)
4. State clearly the answer you obtained for problem #3. Based on this number (even if incorrect!), state whether your answer means that the plant population dies out, increases without limit, or becomes stable.

 

[stapel]

Pn=(2/3)pn-1 + (1/3)pn-2

I will guess that "P" and "p" are actually meant (contrary to mathematical standards) to be the same thing. I will guess that grouping symbols and subscripting were omitted. What you've posted means this:

. . . . .\(\displaystyle \mbox{a. }\, Pn\, =\, \left(\dfrac{2}{3}\right)pn\, -\, 1\, +\, \left(\dfrac{1}{3}\right)pn\, -\, 2\)

I'll be guessing that you actually meant this:

. . . . .\(\displaystyle \mbox{b. }\, P_n\, =\, \dfrac{2}{3}\, P_{n-1}\, +\, \dfrac{1}{3}P_{n-2}\)

But what is the point of this equation? Does it relate to what follows? If so, how? Also, what were the "seed" values?

Now suppose there were 90 plants the first year and 45 plants the second year.

"Now" suggests that text or other information preceded the above. What was that information? What changed in the "now" part?

1. How many plants are there in the third year? (my answer: 60)
2. How many plants are there in the fourth year? (my answer: 55)

How did you obtain these values?

3. Using the recursion relation Fn=F n-1 + F n-2 , we discovered that the limit ratio of the Fibonacci sequence was one half of one plus square root of five, which is approximately 1.6.

How did "we" do this? Where did this happen? Is this just supposed to be assumed for this part of the exercise? Do the Fn's relate at all to the Pn's? If so, how?

Questions:
(a) If the limit ratio had been 1⁄2, what could you say about the numbers far out in the sequence?
(b) What if the limit ratio was 1? 3. Find the limit ratio for the sequence. (Same as problem 38 in the Homework with different numbers; you should mimic what was done in class to get the golden ratio – the limit ratio for the Fibonacci sequence.)

What was "problem 38"? What have you done, modelling on that example? Also, is this exercise (3)?'

Please be complete. Thank you!