Peter gave you sound advice.
No brood cycles are not expedential. they take the same amount of time each cycle.
However, Once the hive begins building the growth is exponential as nurse bees are required to cover the brood. The more nurse bees the more eggs the queen can lay which results in more nurse bees, Then allowing the queen to lay even grater numbers of eggs.
Although your season is short, you still have time for this hive to build up and prepare for winter. Feed the heck out of that hive, including pollen patties just feed them a little at a time so as not to promote hive beetles.
Beekeeping should not be stressful, If it is them you are either not properly prepared, or you allow your concern for the bees to effect you too deeply. we all handle difficulties to different degrees. always keep in mind some things are beyond your control, there is usually a solution to any difficulty, Don't sweat the small stuff. and the most important rule of all. It's all small stuff!
