These are formulas for predicting both the power and cost progression of upgrades determined in September 2012. The formulas can be found here.
Note: while the current formula used to predict a weapon's upgrade powers uses its Lvl 1 power, it now appears that it's more accurate to use its Lvl 8 power to retrospectively determine its previous power levels (this appears to cause less rounding errors, as demonstrated from the power levels of the Xmax-Tree introduced in Version 2.20).
The Incremental Costs tells you the total amount of gold you'll need to spend to fully upgrade a given level from each of its upgrade levels, which is why the costs decrease as the levels increase -- this is not an error.
Upgrade Formulas
Costs
Level 1: (Purchase Cost)
Level 2: Purchase Cost x 0.3
Level 3: Purchase Cost x 0.5
Level 4: Purchase Cost x 0.7
Level 5: Purchase Cost x 1.0
Level 6: Purchase Cost x 2.0
Level 7: Purchase Cost x 3.0
Level 8: Purchase Cost x 4.5
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TOTAL: Purchase cost x 12
Upgrades
Level 1: (Initial Power)
Level 2: Initial Power x 1.15
Level 3: Initial Power x 1.35
Level 4: Initial Power x 1.60
Level 5: Initial Power x 1.90
Level 6: Initial Power x 2.30
Level 7: Initial Power x 2.80
Level 8: Initial Power x 3.50
