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Twenty deposits (Posted on 2016-01-04) Difficulty: 3 of 5
A Texas oilman who was an amateur number theorist opened a new bank account by depositing a certain integral number of dollars, which we shall call x.
His second deposit, y, also was an integral number of dollars.
Thereafter each deposit was the sum of the two previous deposits.
(In other words, his deposits formed a generalized Fibonacci series.)
His 20th deposit was exactly a million dollars.

What are the values of x and y, his first two deposits?

Source: problem by Leonard A. Monzert

No Solution Yet Submitted by Ady TZIDON    
Rating: 3.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution solution | Comment 1 of 4
Dividing 1,000,000 by phi^18, rounded gives 173; dividing by phi again, rounded gives 107. But starting with 107 and 173 results in a 20th deposit of only 999,801.

Playing around by fiddling with the two numbers 107 and 173 doesn't seem to produce the desired 1,000,000, so a different tack is needed:

Working the other way around, assume the 19th deposit is 1,000,000 / phi, rounded, or  618034. Then work backwards, subtracting  618034 from 1,000,000, etc., each time subtracting the newly found value from the previously found value.

Working backward, via subtraction, from the 18th to the 1st deposit:

 381966 
 236068 
 145898 
 90170 
 55728 
 34442 
 21286 
 13156 
 8130 
 5026 
 3104 
 1922 
 1182 
 740 
 442 
 298 
 144 
 154 

So the first two deposits were $154 and $144 respectively.

BTW, doing the same while assuming a 19th deposit of merely one dollar more or less away from  618,034 would result in a couple of negative values along the way, so this solution is it, exactly.


DEFDBL A-Z
CLS
OPEN "20dep.txt" FOR OUTPUT AS #2
c = 1000000
b = 618034
FOR n = 18 TO 1 STEP -1
   a = c - b: PRINT #2, a
   c = b: b = a
NEXT n
CLOSE 2


  Posted by Charlie on 2016-01-04 15:33:04
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