I don't think your equation is right, JK. Per your data:
| Number of Players | Coin Cost |
|-------------------|-----------|
| 50 | 250 |
| 75 | 450 |
| 100 | 700 |
Your equation would expect 250 for 50, 375 for 75, and 500 for 100. The cost instead scales superlinearly with number of players. It's probably some progressive cost system where each additional player (or player bracket) costs more than the last.