Question

The total reserves of a non-renewable resource are 400 million tons. Annual consumption is 25 million tons per year and increases at 1% per year. After how many years will reserves be exhausted?

Do I start by stating the Q_0 = 400 for year 0. Q_1 is 400-25(1.01), and year 2 is 400-25(1.01)^2 ?? Is this right or even close to it?

Answer 1

I think you are on the right track... I need to review this to be sure, but I think that is the right idea.

Answer 2

Ok so, I think my general term for this should be \[Q _{n}=400-25(1.01)^n\]

or, should it be the above but with n25(1.01)^n ?

Answer 3

I think the first one is correct... I'm trying to review at the same time here :)

Do you know how to do logarithms? Your expression is good, but I didn't know whether you were going to solve by trying successive "n" values or solve analytically with logs.

Answer 4

Mmm, this section is in finite and infinite geomentric series. So I think I need to approach it in finding a general term, and the solve for when Q will equal 0 (probably with logs).

Answer 5

that definitely sounds like the right way to go.

Answer 6

Thanks, I just cant seem to figure out an expression that will do this recursively. I think my first expression is close, but It is missing something.

Answer 7

I know the frustration... if I can work it out, I will post it :)

Answer 8

oooh, what if instead of 400 i use Qn-1 so \[Q _{n}=[Q_{n-1}-25(1.01)^n\]

Answer 9

this is tough to tell from the problem, but I was thinking that in year 1, you decline by 25*1^0... in other words, year 0 has 400, year 1 has decline of 25 to be 375.

But you could read it that the decline at the initial point is already 25, so year 1 declines by 25(1.01)

Answer 10

The exponent is whats making this tricky. I think if I say \[Q _{n}=Q _{n-1}-25(1.01)^n\] for \[n \ge1\] then I am safe from going before year 0. And then it should work no matter what the year. Then I just solve for zero.

Answer 11

Anyhow, thank you for the help, I will leave this open, but I have to go.

Answer 12

Good luck... sorry I didn't have a ready-to-post explanation, but I will watch for other responses too... need to know this stuff! :)

Answer 13

Need help determining whether this series converges or not: \[\sum_{1}^{\infty} \frac{ (2n)! }{ (n!)^2 }\] My thoughts are this is a candidate for ratio test, so I rewrote it \[\left| \frac{ (2(n+1))! }{ ((n+1)!)^2 } \right| * \left| \frac{ n!^2 }{ (2n)! } \right|\] But I do not know how to simplify this expression.