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Here's the question:

A population of weasels grows with rate $3\%$. We define $w(k)$ to be the number of weasels after $k$ years from now, and the current number of weasels is $350$.
a) Find the difference equation that describes the population growth in year $k$ for $k=1,2,...$
b) Solve the difference equation you found in (a) (find an analytical formula for $w(k))$.
c) In how many years would we have $700$ weasels?

My solution:
a) $w(k)=1.03w(k-1)$. (Side question, does it matter if i do $w(k+1)=1.03w(k)$?).


And here the fun starts:
b) I've first did $w(0)=350$, and calculated it as $350*1.03^k$, but the provided answer was $w(k)=339.8*1.03^k$.
And that comes only from the calculation $w(1)=w(0)^1.03\Longrightarrow w(0)=339.8$.
So from this question and the provided answer, I got that $w(0)$

And here I got really confused:
c) $339.8*1.03^k=700\Longrightarrow k=24.44$ which means we need $25$ years atleast to get $700$ weasels.
And the provided answer for (c) was $24$, and that got me really confused, it can be done by using $350*1.03^k=700$.

Was I supposed to understand that I need to calculate $w(0)$?
Why if I use $w(0)$ in (c), I would get a wrong answer?


The point of me asking this question is to see if this question is defined correctly, or I'm making mistakes and confusing myself.
Thanks in advance.

Pwaol
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1 Answers1

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Your answers for (a) and (b) are correct, and their answer for (b) is wrong. Some people would write $w(k)=350*1.03^{k}$, like you did, while some feel the need to do these problems as $w(k) = 339.8 * 1.03^{k+1}$, which should still give $w(0)=350$. They seem to have gotten distracted with an answer somewhere between these solutions.

I would also agree that something is suspicious on (c), but we can't say what. Either they went back to the correct solution for (b), and found the solution that you would (24 years), or they did the same calculation that you did, and rounded down from 24.44 to 24 (which might happen in a lower level textbook, but is not a reasonable answer).

Either way, your difficulties seem to be caused by how the question is written, not in any misunderstandings of the material.

Vaekor
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  • Thank you, I got a little question, if it was provided that $k=1,2,3...$ at the start just like now, and in (b) they asked for $k=0,1,2,...$ (They didn't) but lets say they did, Then I would need to do what I did and find $w(0)$? – Pwaol Nov 02 '21 at 18:03
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    w(0)=350 is stated in the problem because they said that k is years from now (so now is k=0) and there are currently 350 weasels (so w(0)=350) – Vaekor Nov 02 '21 at 18:29