Solving for # of periods in geometric progression

[mand1022]

Need some help on a geometric progression problem to solve for the number of periods. Here is the problem: A rabbit farm has 2000 rabbits in the first period. Each month 20 rabbits are added by the game warden. In addition, the population grows by 3% per month by reproduction. How many months does it take for the population to reach 5000? I think I got close to solving this, but...could use some help

[nona.m.nona]

I think I got close to solving this, but...could use some help

Please reply with your efforts thus far, so it may be determined where you are needing assistance. Thank you.

[mand1022]

I was using the following formula:

Sn = a[(1-r^n)/(1-r)] where a is a for the first period i.e. 2000 rabbits in this case. 5000 is Sn as it is the total rabbits at the end number of periods. The goal is to solve for n or the number of periods required to get to 5000. r is the rate of increase or 3% so 1.03. So I get to the following:

5000 = 2000[(1-1.03^n)/(1-1.03)] + 20n The 20 n represents the 20 rabbits added each period. My problem now is to solve for n i.e the number of periods.

I assume this is the right formula to go about solving the problem, just can't remember how to solve, guessing I need a log?

[nona.m.nona]

So I get to the following:

5000 = 2000[(1-1.03^n)/(1-1.03)] + 20n The 20 n represents the 20 rabbits added each period.

According to your formula, it would appear that the rabbits which are added in are assumed not to reproduce. Is this a reasonable assumption?

[mand1022]

No, the population of rabbits starts at 2000 and 20 are added every month and each of the rabbits reproduces by 3% every month. I would assume that the rabbits added are reproducing each month as well I just cannot develop a proper formula that will allow me to figure out how many months it takes all the rabbits to have a population of 5000. My first two posts were the formula I was working on, but I am not sure if that is correct and, if it is, I am not sure how to work it to yield me an answer. (months are the variable n)

[nona.m.nona]

NI would assume that the rabbits added are reproducing each month as well I just cannot develop a proper formula

When in doubt as to how to develop a modeling formula, it is frequently useful to show one's work explicitly.

At the start (at step 0), the population is 2 000.

At the one-month mark, the population is regarded as having grown by three percent: 1.03 * 2 000.
Also, twenty more units are added to the population: 1.03 * 2 000 + 20

At the two-month mark, the population is regarded as having grown by three percent: 1.03(1.03 * 2 000 + 20)
Also, twenty more units are added to the population: 1.03(1.03 * 2 000 + 20) + 20
Simplifying yields: 1.03² * 2 000 + 1.03 * 20 + 20

Follow the pattern for the three-month mark. Simplify. And again for the four-month mark. Continue until you are able to extract a generalized formula for the n-th-month mark.