Question 1171556
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I take along a bottle with some wine in it for an excursion in the spring. 
On reaching a tavern, I double the bottle’s content and drink 1 9/10 dou in the tavern. 
After passing 4 taverns, the bottle is empty. How much wine was there at the beginning? 
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            My interpretation of this problem is different from that of @ankor@dixie-net.com.


            THEREFORE,  my solution and my answer are different.



<pre>
We have a sequence of numbers  {{{x[0]}}}, {{{x[1]}}}, {{{x[2]}}}, {{{x[3]}}}, {{{x[4]}}}.


{{{x[0]}}} is the amount of wine in the bottle originally.

{{{x[k]}}} is the amount of wine in the bottle at the Exit     from the  k-th tavern 
                                (same as at the entering to   the (k+1)-th tavern), k = 1,2,3,4.


From the problem's condition, we have this recursive formula

    {{{x[k+1]}}} = {{{2*x[k]-1.9}}},      (1)


which is the formal mathematical translation of the problem's description.


We are given that  {{{x[4]}}} = 0, and they want we find {{{x[0]}}}.


Surely, the problem should be solved in a backward manner.

For it, I express {{{x[k]}}} from (1) in this way

    {{{x[k]}}} = {{{(x[k+1]+1.9)/2}}}    (2)


Now I will go step by step BACKWARD from  k= 3  with  {{{x[4]}}} = 0.


In order for to facilitate my calculations, I use MS Excel in my computer with implemented formula (2) in it.


It provides all calculations in seconds, giving me THIS Table below

    k         {{{x[k]}}}

--------------------------

    4		0
    3		0.95
    2		1.425
    1		1.6625
    0		1.78125


So, the <U>ANSWER</U> is: the initial amount of wine in the bottle was  1.78125  units of the volume.
</pre>

SOLVED.