Lesson Entertainment problem: Uninterrupted withdrawing money from a retirement fund

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<H2>Entertainment problem: Uninterrupted withdrawing money from a retirement fund</H2>

<H3>Problem 1</H3>Suppose a man is &nbsp;25 &nbsp;years old and would like to retire at age &nbsp;60. &nbsp;Furthermore, &nbsp;he would like to have a retirement fund 
from which he can draw an income of &nbsp;$100,000 &nbsp;per year&nbsp;--&nbsp;forever&nbsp;! &nbsp;&nbsp;How can he do it? &nbsp;Assume a constant &nbsp;APR &nbsp;of &nbsp;8%.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;From the first glance, &nbsp;the idea of this problem seems to be insane; &nbsp;but actually, &nbsp;it is absolutely realistic,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;at least, &nbsp;mathematically :)

<B>Solution</B>

<pre>
First idea that comes to mind is to have infinitely many money at the account.

The second idea is to have a printer and print money.

Third idea is even better than the two previous. 
Indeed, all this person needs is to have so much money in his account that 8% of it is more than $100,000, i.e.

0.08*X >= 100000,   which implies   X >= {{{100000/0.08}}} = 1,250,000 dollars.

Then each year he can withdraw $100,000 (which is 8% of $1,250,000) at the beginning of the year and spend this money for living; 
at the end of the year, at the compounding step, the bank will return  equal or even greater amount than 0.08*1250000 = 100,000 dollars, 
so his balance will be stable or will increase from year to year.

It is a rough estimation.

We can consider more realistic case, when he withdraws $100,000 at the first day of the year and spends it for living; at the end of the year,

the bank compounds 8% of the remaining amount; it imposes THIS inequality for the unknown starting amount X

0.08*(X-100000) >= 100000,

which gives

0.08X - 0.08*100000 >= 100000,

0.08X > (1+0.08)*100000

x >= {{{((1+0.08)*100000)/0.08}}} = 1350000.

<U>Answer</U>.  Having  $1,350,000 or more initially at the account provides the sought condition.
</pre>

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