Why these two methods to get the interest earned on recurring deposits return different results?

Question

Suppose that at the beginning of each month I make a deposit of 100 dollars on an account with a 1% monthly rate. I want to calculate the earned interest after 6 months.

Method 1

End of month Earned Interest 1 1 (100*0.01) 2 2.01 ((200 + 1)*0.01) 3 3.0201 ((300 + 2.01)*0.01) 4 4.030201 ((400 + 3.0201)*0.01) 5 5.04030201 ((500 + 4.030201)*0.01) 6 6.0504030201 ((600 + 5.04030201)*0.01)

Total earned interest after 6 months:
21.1510060301

Method 2

End of month Account Balance (invested + interests) 1 101 (100*1.01) 2 203.01 ((101+100)*1.01) 3 306.0401 ((203.01+100)*1.01) 4 410.100501 ((306.0401+100)*1.01) 5 515.20150601 ((410.100501+100)*1.01) 6 621.3535210701 ((515.20150601+100)*1.01)

Total earned interest after 6 months:
21.3535210701 (621.3535210701 - 600)

Why the difference?

With the second method, more than 20 cents more are earned. I thought the difference could be because of rounding issues, but I've even tried to write a Clojure program using its Ratio type and the differences are the same:

211510060301/10000000000 (method 1)
213535210701/10000000000 (method 2)

So, how is the difference explained?

Answer 1

Your method 1 is incorrect. You forgot to add the first dollar of interest to the amount on which interest was calcuated in month 3 (and the same error is repeated thereafter). Here is the correct version:

End of month    Earned Interest
1               1            (100*0.01)
2               2.01         ((200 + 1)*0.01)
3               3.0301       ((300 + 1+2.01)*0.01)

A better table might have a column for the cumulative interest and account balance:

End of month   Interest in that month    Cumulative interest   Account balance before interest
1               1            (100*0.01)           1                 100
2               2.01         (201*0.01)           3.01              201
3               3.0301       (303.01*0.01)        6.0401            303.01