Help with recurrence relations question
Posted 21 October 2004 - 05:17 PM
of the harmful
''Aquatec''was found to kill 98 percent of the harmful Bacteria present but another 2450 new Bacteria appeared daily.
''Magiclean'' was found to kill 96 present but another 2400 new Bacteria appeared daily.
Compare the effectiveness, in the long-term, of these two products as far as the eradication of harmful bacteria is concerned.
Posted 21 October 2004 - 05:49 PM
an+1 = 0.02an + 2450
Since -1 < 0.02 < 1 there is a limit.
Let an+1 = mn = L
L = 0.02L + 2450
L - 0.02L = 2450
0.98L = 2450
L = 2450/0.98
L = ...
mn+1 = 0.04mn + 2400
Since -1 < 0.04 < 1 there is a limit.
Let mn+1 = mn = L
L = 0.04L + 2400
L - 0.04L = 2400
0.96L = 2400
L = 2400/0.96
L = ...
Since [insert name of better cleaner] converges at a lower value (and thus keeps the amount of bacteria at a lower amount), this is a more effective cleaner than [insert name of poorer cleaner]
Posted 21 October 2004 - 09:37 PM
Posted 21 October 2004 - 11:31 PM
Come to think of it...there wasn't a recurrence relations question in the 2004 exam.
Posted 21 November 2004 - 05:48 PM
Posted 22 November 2004 - 10:50 PM
Basically what you're doing it finding a formula for the number of bacteria.
With that there will be a limit if the value you multiply the original by is between 1 and -1. This is because eventually there will be a value by which the number you multiply by added to whatever random number afterwards will be the same as the previous and won't go any higher.
With this you find the limit. Since at the limit, the initial u value will equal that of the next, we can let both u's be the same, ie. call it L. Then it's just a case of basic algebra to get the limit L.
For the most effective cleaner here, it'll be the one that allows for less bacteria to live.
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