# Geometric Series to Solve for Year a Resource Will Be Depleted

The original problem is as follows:

A community has 300 million tons of a non-renewable resource. Annual consumption is 25 million tons per year. Consumption is expected to decrease by 10% each year. Will the resource ever be depleted?

I set up the geometric series as

Total usage = 25[1 + (0.90) + (0.90)^2 + ...]
= 25[1/1-0.90]
= 250 million tons
No, the resource will not be depleted at this rate.


The second part of the problem asks:

What is the minimum percentage they can decrease consumption by to guarantee the resource does not run out?

Total usage = 25[1 + x + x^2 + ...]
= 25[1/1-x] = 300
x = 0.916
Minimum we can decrease consumption by is 8.4%


The final part of the question asks:

Suppose that it is not possible to decrease consumption of this resource by the previously given amount. The population is only able to decrease consumption by 5% each year. After how many years will the resource run out?

But I'm not exactly sure how to set this up. I'm assuming that solving for the year is just solving for n in the series where the sum up to (0.95)^n = 300, but I'm not sure how to go about setting up this equation.

Welcome on math.stackexchange! For the geometric series there is not only a formula for $$\sum_{n=0}^\infty q^n$$ but also for the finite sums, namely $$\sum_{n=0}^k q^n=\frac{1-q^{k+1}}{1-q}$$ and this holds for $$q \not =1$$. With this formula you can then calculate the year.