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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)

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Solution Solution | Comment 3 of 4 |
His subsequent deposits are X+Y, X+2Y, 2X+3Y, ...
The general pattern is that the nth deposit equals F(n-2)*X + F(n-1)*Y, where F(n) is the nth Fibonacci number.

The 1,000,000 dollar 20th deposit then equals F(18)*X + F(19)*Y = 2584*X + 4181*Y.
Now the solution is a positive integer solution to 2584*X + 4181*Y = 1,000,000.

Fibonacci numbers obey the rule F(2n-1)*F(2n+1) - F(2n)^2 = 1.  
Then X=-F(18)=-2584, Y=F(17)=1597 is a solution to 2584*X + 4181*Y = 1.
Multiplying this particular solution by 1,000,000 yields X=-2,584,000,000, Y=1,597,000,000.
Generalizing this to all solutions with parameter Z: X=-2,584,000,000+4181*Z, Y=1,597,000,000-2584*Z

A positive integer solution occurs when Z causes both X and Y to be greater than 0.
-2,584,000,000+4181*Z > 0 implies Z>618033.96
1,597,000,000-2584*Z > 0 implies Z<618034.06
There is only one integer Z which satisfies both inequalities: Z=618034

Substituting this value of Z yields X=154, Y=144.  The first two deposits were $154 and $144 in that order.

Edited on January 5, 2016, 12:19 pm
  Posted by Brian Smith on 2016-01-05 12:18:08

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