Sequence - Gas reserves | Practice question

[Eric]

Hey,

I messed up 25% of my last exam because I missed the concept behind geometric and arithmetic series. Next week is my resit and I have spent quite some hours trying to understand the concept and logic behind these series but I didn't succeed yet. There's this one question on an example exam which I just can't solve. It feels quite different from the ordinary question. It is not really the answer I am interested in, but the logic behind it. It would be great if some explanation could be provided along each relevant step so I can re-construct similar questions and finally fully understand the concept.

The question (this is what I can remember of it, the layout was similar but the numbers were different):

Assume there is a country with gas reserves of 1 billion tonnes, each year a total of 16 million tonnes is used.
a). How many years will it take before the gas reserves are depleted?

b). Suppose the government enacts a policy which reduces the usage of gas by 0,5% yearly. How many years before the gas reserves are depleted?(My interpretation of the question: After the first year 99,5% * 16 million is used, after the second year 99% * 16 million is used etc.)

c). Suppose the government enacts a similar policy but instead it reduces the usage of gas by 1% yearly. How many years before the gas reserves are depleted if this is the case?

Question a). is easy, but for b). and c). I can't seem to find a good approach to the question. I have literally tried up to a dozen of arithmetic and geometric approaches but I seem to fail every single time.

Thanks for the effort!

Eric

 

[Ishuda]

Hey,

I messed up 25% of my last exam because I missed the concept behind geometric and arithmetic series. Next week is my resit and I have spent quite some hours trying to understand the concept and logic behind these series but I didn't succeed yet. There's this one question on an example exam which I just can't solve. It feels quite different from the ordinary question. It is not really the answer I am interested in, but the logic behind it. It would be great if some explanation could be provided along each relevant step so I can re-construct similar questions and finally fully understand the concept.

The question (this is what I can remember of it, the layout was similar but the numbers were different):

Assume there is a country with gas reserves of 1 billion tonnes, each year a total of 16 million tonnes is used.
a). How many years will it take before the gas reserves are depleted?

b). Suppose the government enacts a policy which reduces the usage of gas by 0,5% yearly. How many years before the gas reserves are depleted?(My interpretation of the question: After the first year 99,5% * 16 million is used, after the second year 99% * 16 million is used etc.)

c). Suppose the government enacts a similar policy but instead it reduces the usage of gas by 1% yearly. How many years before the gas reserves are depleted if this is the case?

Question a). is easy, but for b). and c). I can't seem to find a good approach to the question. I have literally tried up to a dozen of arithmetic and geometric approaches but I seem to fail every single time.

Thanks for the effort!

Eric

First, if you can do (b), you should be able to do (c) since the problems are 'the same' but with a different amount reduction. So, let's look at (b)
b). Suppose the government enacts a policy which reduces the usage of gas by 0,5% yearly. How many years before the gas reserves are depleted?(My interpretation of the question: After the first year 99,5% * 16 million is used, after the second year 99% * 16 million is used etc.)

You interpretation is one of two which could be correct and is an arithmetic progression if it said 'reduces the usage of gas by 0,5% of the original amount yearly'. That is the difference between neighbouring years are the same
\(\displaystyle \begin{matrix}Beginning\, Year & Amount\, Used & Difference\\
0 & 1.000 * 16M & - \\
1 & 0.995 * 16M & -0.005 * 16M = Year\, 1 - Year\, 0 \\
2 & 0.990 * 16M & -0.005 * 16M = Year\, 2 - Year\, 1 \\
3 & 0.985 * 16M & -0.005 * 16M = Year\, 3 - Year\, 2 \\
...& & \end {matrix}\)

However, IMO, what is generally meant by the way the problem is stated is as a geometric progression and it is understood that the problem is to be interpreted as 'reduces the usage of gas by 0,5% of the last years amount yearly'. That is the ratio between neighbouring years are the same
\(\displaystyle \begin{matrix}Beginning\, Year & Amount\, Used & Ratio\\
0 & 0.995^0 * 16M & - \\
1 & 0.995^1 * 16M & .995 = \dfrac{Year\, 1}{Year\, 0} \\
2 & 0.995^2 * 16M & .995 = \dfrac{Year\, 2}{Year\, 1} \\
3 & 0.995^3 * 16M & .995 = \dfrac{Year\, 3}{Year\, 2} \\
...& & \end {matrix}\)

You should, again IMO, find out which interpretation is meant by your instructor.

 

[stapel]

I messed up 25% of my last exam because I missed the concept behind geometric and arithmetic series. Next week is my resit and I have spent quite some hours trying to understand the concept and logic behind these series but I didn't succeed yet.

For general instruction (before attempting specific exercises), try online lessons, such as here.