Just add one dollar to that first payment, and you will be ahead by a million dollars in 20 years.
Explanation -- how it works:
After the first year:
Case A (without paying the extra dollar). You pay a million dollars, and still owe a million principal.
Case B. You pay $1,000,001, and still owe $999,999 principal. You are ahead by $1 (compared to case A).
After the second year:
Case A. You pay a million dollars, and owe a million principal (just like after the first year).
Case B. You pay a million dollars, and owe $999,998 principal. You are ahead by $2.
After the third year:
A. (same as above)
B. You pay a million dollars, and owe $999,996 principal. You are ahead by $4.
After the 4th year:
A. (same as above)
B. You pay a million dollars, and owe $999,992 principal. You are ahead by $8.
Notice that the amount you are ahead doubles every year. So far the amounts are small. But if you start with $1 and double it 20 times, you end up with $1,048,576 -- more than needed to pay off all of the $1,000,000 principal. So you end up having paid back the whole debt.
That one dollar investment paid you back a million dollars in 20 years.
Variation: Suppose that with your first repayment of $1,000,000, you came up with an extra thousand dollars -- just an extra tenth of 1% of what you had to pay anyway. Then after 10 years, you would be out of debt. That's because if you start with $1,000 and double it 10 times, you end up with $1,024,000 -- more than enough to pay off the principal. That $1,000 investment puts you a million dollars ahead after 10 years.
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