Problem 1
To begin with, Aldi has 0 mg of the medicine in his bloodstream.
Then Haidar gives her dog 150 mg of the medicine.
The day progresses along and at the end of the day, 40% of the medicine remains. So 0.40*150 = 60 mg remains.
At the start of day 2, Haidar gives another 150 mg dose. The 60 mg bumps up to 60+150 = 210 mg. This becomes 0.40*210 = 84 mg at the end of day 2.
At the start of day 3, she gives her dog yet another 150 mg dose. So the 84 increases to 84+150 = 234. This decays to 0.40*234 = 93.6 mg at the end of day 3.
Finally, at the start of day 4, the 150 mg dose bumps the 93.6 up to 93.6+150 = 243.6 mg, which then decays to 0.40*243.6 = 97.44 mg.
Here's a table to summarize everything
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Problem 2
The first term is to indicate the dog has 0 mg of medicine in his bloodstream to start off with.
Then we add on 150 mg to that amount, and take 40% of the sum.
This indicates that
Similarly,
and so on.
In general, we have this recursive definition
The refers to the nth dose while
is the dose amount just before the nth dose.
This sequence is neither arithmetic nor geometric.
It's not arithmetic because we aren't adding the same number to each term to get the next term. It's not geometric because we aren't applying the same common ratio to multiply from term to term.
The mix of "plus 150" and "times 0.40" is almost like this is a hybrid of arithmetic and geometric respectively. However, it's not purely one or the other.