Hello! I need help finding an equation that takes two variables.

Billy has

It looks a lot like a geometric sum, but sometimes it's less. Here's the equation I've been using so far:

Start with sum of geometric series: (1−m^(Log(n)/Log(m)))÷(1−m)

Simplify:

\(\displaystyle \frac{1-n}{1-m}\)

With m=2 this works.

But with m>2 it doesn't always. m=3 (values of n where my equation works are in bold):

And m=4...

**Problem in word form.**Billy has

*n*game tokens to play arcade machines that take 1 game token each. After playing*m*games, the arcade gives him 1 game token back. How many total games*t*can Billy play with an inital*n*game tokens?**Work done so far**It looks a lot like a geometric sum, but sometimes it's less. Here's the equation I've been using so far:

Start with sum of geometric series: (1−m^(Log(n)/Log(m)))÷(1−m)

Simplify:

\(\displaystyle \frac{1-n}{1-m}\)

With m=2 this works.

n | m | t-n | (1-n)/(1-m) |

0 | 2 | 0 | -1 |

1 | 2 | 0 | 0 |

2 | 2 | 1 | 1 |

3 | 2 | 2 | 2 |

4 | 2 | 3 | 3 |

5 | 2 | 4 | 4 |

6 | 2 | 5 | 5 |

7 | 2 | 6 | 6 |

8 | 2 | 7 | 7 |

9 | 2 | 8 | 8 |

10 | 2 | 9 | 9 |

n | m | t-n | (1-n)/(1-m) | Difference |

0 | 3 | 0 | -0.5 | -0.5 |

1 | 3 | 0 | 0 | 0 |

2 | 3 | 0 | 0.5 | 0.5 |

3 | 3 | 1 | 1 | 0 |

4 | 3 | 1 | 1.5 | 0.5 |

5 | 3 | 1 | 2 | 1 |

6 | 3 | 2 | 2.5 | 0.5 |

7 | 3 | 3 | 3 | 0 |

8 | 3 | 3 | 3.5 | 0.5 |

9 | 3 | 4 | 4 | 0 |

10 | 3 | 4 | 4.5 | 0.5 |

11 | 3 | 4 | 5 | 1 |

12 | 3 | 5 | 5.5 | 0.5 |

13 | 3 | 5 | 6 | 1 |

14 | 3 | 5 | 6.5 | 1.5 |

15 | 3 | 6 | 7 | 1 |

16 | 3 | 7 | 7.5 | 0.5 |

17 | 3 | 7 | 8 | 1 |

18 | 3 | 8 | 8.5 | 0.5 |

19 | 3 | 9 | 9 | 0 |

20 | 3 | 9 | 9.5 | 0.5 |

27 | 3 | 13 | 13 | 0 |

81 | 3 | 40 | 40 | 0 |

n | m | t-n | (1-n)/(1-m) | Difference |

0 | 4 | 0 | -0.33 | -0.33 |

1 | 4 | 0 | 0.00 | 0.00 |

2 | 4 | 0 | 0.33 | 0.33 |

3 | 4 | 0 | 0.67 | 0.67 |

4 | 4 | 1 | 1.00 | 0.00 |

5 | 4 | 1 | 1.33 | 0.33 |

6 | 4 | 1 | 1.67 | 0.67 |

7 | 4 | 1 | 2.00 | 1.00 |

8 | 4 | 2 | 2.33 | 0.33 |

9 | 4 | 2 | 2.67 | 0.67 |

10 | 4 | 2 | 3.00 | 1.00 |

11 | 4 | 2 | 3.33 | 1.33 |

12 | 4 | 3 | 3.67 | 0.67 |

13 | 4 | 4 | 4.00 | 0.00 |

14 | 4 | 4 | 4.33 | 0.33 |

15 | 4 | 4 | 4.67 | 0.67 |

16 | 4 | 5 | 5.00 | 0.00 |

17 | 4 | 5 | 5.33 | 0.33 |

18 | 4 | 5 | 5.67 | 0.67 |

19 | 4 | 5 | 6.00 | 1.00 |

20 | 4 | 6 | 6.33 | 0.33 |

64 | 4 | 21 | 21 | 0 |

256 | 4 | 85 | 85 | 0 |

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