I'm preparing for my mathematics exam and I am stuck on something which I believe should be simple. The question is as follows:
A rough estimate of the total oil and gas reserves of a country at the beginning of 2000 was 12 billion ($12\cdot 10^9$) tons. Production that year was approximately 50 million tons. When will the reserves be exhausted if production is kept at the same level?
My progression: I believe, correct me if I'm wrong, that I am dealing with a geometric sequence here. Then the formula (proof included in the book) is given as: $S_n=a_0\cdot\frac{1-r^{N+1}}{1-r}$. I believe that in this case it is merely N that is not given, hence I should be able to solve it through this formula. Therefore, I state the following: $S_n=12\cdot 10^9$, $a_0=50\cdot 10^6$ and $r=50\cdot 10^6$.
Now I do know that the answer should be 240 years. However, I already come to this answer when I calculated $S_n/a_0$, which I cannot understand. My idea was to solve as follows: \begin{align*} S_n=a_0\frac{1-r^{N+1}}{1-r} \implies \frac{S_n}{a_0}\cdot (1-r)-1=-r^{N+1}\\ \implies -\left(\frac{S_n}{a_0}\cdot (1-r)-1\right)=r^{N+1} \implies ln\left(-(\frac{S_n}{a_0}\cdot (1-r)-1)\right) =(N+1)ln(r) \implies \frac{ln(-(\frac{S_n}{a_0}\cdot (1-r)-1-1))}{ln(r)}-1=N \end{align*}
Perhaps some of you are laughing, cause it might look ridiculous, but to me it looks pretty logical. Obviously, the answer (N) is incorrect, as it is lower than 1 (although with the use of $ln$ it doesn't really come too much as a surprise). Please note that the negative sign in $ln$ yields a positive number, hence I think that I may use it.
Long story short, the questions are: 1) Am I correctly identifying the symbols? 2) Why does $S_n/a_0$ already yield the correct answer? 3) Why would the method not work?
Hopefully I am not asking too much. Thank you in advance.
It's not a geometric sequence.
It's an arithmetic sequence, but can be dealt with even more simply than that.
You start with $12,000,000,000$ tons of oil in your reserves.
Each year you pump $50,000,000$ tons out of the ground.
Your reserves are therefore dropping by $50,000,000$ tons each year - an arithmetic sequence.
You will have used up all the reserves in $12,000,000,000 \over 50,000,000$ years $= 240$ years.