Word problem on sequences help

[Leilau]

Given an animal’s population is 4,000,000 in 1989 2,520,000 in 1990 and 1,587,600 in 1991.
If this trend continues and the population falls below 200,000 it will be placed on the list of
endangered. Write a recursive function for this situation. In what year would it be placed on the list
of endangered species?
So far I’m having a problem on the recursive formula i have it as u(n-l)x(1-0.63) and U of 0 might be 4,000,000. That’s what I’ve gotten so far and you need a graphic calculator to help solve this.

 

[ksdhart2]

I think you're mostly on the right track, but it's hard for me to tell for sure. The very first thing I notice is that you need to be a lot more clear about what steps you're taking and what each variable you introduce really stands for. In your working so far you have four variables (\(u\), \(n\), \(l\), and \(x\)), but it's not particularly clear to me what most of them stand for - the only thing I can gather is that you're using \(u(t)\) as the main function, to give you the population at a certain year.

You say that you think \(U(0)\) might be 4,000,000 [Side note: In math, capitalization is very important. \(u\) and \(U\) are different variables and should never be used interchangeably]. This is certainly a valid way to approach the problem, you just need to follow through and say what the variable \(t\) then represents. Given \(U(0) = 4000000\), we can say that \(U(t)\) represents the animal's population \(t\) years after 1989.

Following up on this, we have \(U(1) = 2520000\) and \(U(2) = 1587600\). Now let's examine the differences between successive terms. What is \(U(0) - U(1)\)? And what is \(U(2) - U(1)\)? Are those values the same? If they are, what type of sequence would that make \(U(t)\), and what do you know about that type of sequences?

If they're not the same, what other types of sequences do you know about? Maybe look at the ratios of successive terms. What is \(\displaystyle \frac{U(1)}{U(0)}\)? And what is \(\displaystyle \frac{U(2)}{U(1)}\)? Are those values the same? If they are, what type of sequence would that make \(U(t)\), and what do you know about that type of sequences?

 

[Dr.Peterson]

Given an animal’s population is 4,000,000 in 1989 2,520,000 in 1990 and 1,587,600 in 1991.
If this trend continues and the population falls below 200,000 it will be placed on the list of
endangered. Write a recursive function for this situation. In what year would it be placed on the list
of endangered species?
So far I’m having a problem on the recursive formula i have it as u(n-l)x(1-0.63) and U of 0 might be 4,000,000. That’s what I’ve gotten so far and you need a graphic calculator to help solve this.

I suppose what you mean is that u(0) = 4,000,000 and, for n>0, u(n) = u(n-1)*(1-0.63), where u is the population and n is the number of years since 1989. If so, you are almost right. You have the right type of sequence.

But why 1-0.63? Did you check this against the given values? You just slipped a little bit.

After correction, this is the first part of the answer.

There are several ways you could answer the second part, one of which is to repeatedly apply the formula, and another is to develop an explicit formula for u(n), using whatever you have learned about this type of sequence, or just thinking about the recursive formula. Either of these can be finished with any kind of calculator. Or you could solve that formula, if you have learned about logarithms.

Maybe you'll need to tell us which of these things you have learned.

 

[Leilau]

I suppose what you mean is that u(0) = 4,000,000 and, for n>0, u(n) = u(n-1)*(1-0.63), where u is the population and n is the number of years since 1989. If so, you are almost right. You have the right type of sequence.

But why 1-0.63? Did you check this against the given values? You just slipped a little bit.

After correction, this is the first part of the answer.

There are several ways you could answer the second part, one of which is to repeatedly apply the formula, and another is to develop an explicit formula for u(n), using whatever you have learned about this type of sequence, or just thinking about the recursive formula. Either of these can be finished with any kind of calculator. Or you could solve that formula, if you have learned about logarithms.

Maybe you'll need to tell us which of these things you have learned.

So I looked at the problem again and I realized my recursive equation is wrong. The common ratio is 0.63 and since the sequence is decreasing the recursive formula is going to be u(n) = u(n-1)*(1-0.37). I put this in the calculator and the answer I got is year 1996 is when it reaches below 200,000. I think maybe that I’m going in the right direction?

 

[ksdhart2]

I agree with your answer that the species becomes endangered in 1996, but if I were your teacher I think I'd like to see a little bit more than just using a calculator. The way I'd go about it is to try and form an explicit definition of the sequence, rather than a recursive one:

• \(U(0) = U(0) \cdot 1 = U(0) \cdot 0.63^0\) [Anything to the \(0^{th}\) power equals 1]
• \(U(1) = U(0) \cdot 0.63 = U(0) \cdot 0.63^1\)
• \(U(2) = U(1) \cdot 0.63 = \left[U(0) \cdot 0.63 \right] \cdot 0.63 = U(0) \cdot 0.63^2\)
• \(U(3) = U(2) \cdot 0.63 = {\color{blue}\big(} \left[U(0) \cdot 0.63 \right] \cdot 0.63 {\color{blue}\big)} \cdot 0.63 = U(0) \cdot 0.63^3 \)
• \( \cdots \)
• \(U(n) = U(0) \cdot 0.63^n = 4000000 \cdot 0.63^n\)

If you sub in 200000 for \(U(n)\) and then solve for \(n\), you'll arrive at the answer.