In a water purification process, water is passed through a largenumber of filter

Question

In a water purification process, water is passed through a largenumber of filters. The first filter removes 1/mth of theimpurities present, where m = 2.10. Each subsequent filter is lessefficient, removing only 1/mth of the amount (say, inkg) that was removed in the previous stage. a) After 7 filters,what is the percentage of impurity remaining in the water,b) If thenumber of filters is increased, what is the minimum percentage ofimpurity than can ever be achieved, and c) What is the bestpossible value of m for this system?

I know that the best value is 2 (part c), but how do I do the firsttwo parts? Can this be modled as a geometric series, in whichcase it can converge to a/1-r if r < 1. How do I figureout what series to model this as?

Explanation / Answer

Let M is the impurities so M/m is the amount of impurities that are removed in the firstfilter M/m^2 is the amount of impurities that are removed in thesecond so on, it is M/m^3 , M/m^4 , etc a/ there are 7 filter so S=M*0,9 So there are only 0,1 percentage of impurities left. b) the series is S=M*(1/m+1/m^2+1/m^3+.....) if the series is increased, so S=M*1/(m-1)=0,909 so the impurities remain percentage is 9%. c) It is seen that if ,=2 then S=M so the impurities remain iszero

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