I have an assignment, which has the following problem: Assume that a student tries to summaries 3 pages of a math script, which has infinite pages.Over night he forgets 2% of his new knowledge. Also, the student has no knowledge whatsoever on the first day. The problem is to show that the recursive formula is a cauchy sequence.
I attempted the following: $(k+3)*98/100$. Now I'm not sure how to show that this formula is a Cauchy sequence.
Assuming that the student $S$ learns the stuff he forgot and some, to reach an equivalent of three pages: Let $n$ be the number of the day and $a_n$ the amount of pages $S$ has memorized after the $n-$th day.
$$a_0=0$$ $$a_1=3$$ On the next day $S$ forgets 2% and learns 3 more pages. So $S$ still knows 98% and learns an additional 3 pages. $$a_2=a_1\cdot0.98+3$$ The same goes for all other days: $$a_3=a_2\cdot0.98+3=(a_1\cdot0.98+3)\cdot0.98+3=3\cdot(0.98^2+0.98+1).$$ In general: $$a_n=a_{n-1}\cdot 0.98+3=3\sum_{k=0}^{n-1}0.98^k.$$
This should help you prove it.