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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.
We have a sequence of numbers , , , , .
is the amount of wine in the bottle originally.
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
= , (1)
which is the formal mathematical translation of the problem's description.
We are given that = 0, and they want we find .
Surely, the problem should be solved in a backward manner.
For it, I express from (1) in this way
= (2)
Now I will go step by step BACKWARD from k= 3 with = 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
--------------------------
4 0
3 0.95
2 1.425
1 1.6625
0 1.78125
So, the ANSWER is: the initial amount of wine in the bottle was 1.78125 units of the volume.
SOLVED.