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

The x component of his 20 deposits are as follows:

x, 0, x, x, 2x, ... x*F(18)

The y component of his 20 deposits are as follows:

0, y, y, 2y, ... y*F(19).

His 20th deposit = x*F(18) + y*F(19) = x*2584 + y*4181 = 1,000,000

x = (1,000,000 - y*4181)/2584 = 387 - 2y + (987y - 8)/2584

Let z = (987y - 8)/2584

Then y = (2584z + 8)/987 = 2z + (610z + 8)/987

We could keep reducing the divisor by creating new variables, but enough already!

Resorting to excel, we find an integer solution at

z = 55

So y = 144

so x = 154

Same result as Charlie