01
Calculate the uncapped sequence
Starting from x at step zero equal to 1, apply the rule x next equals two times x plus 1. Step one is 3, step two is 7, step three is 15 and step four is 31. Each value is calculated from the previous row, not directly from the step number. The sequence contains five displayed states because the initial value is included alongside four updates. A reference table should retain that first row so a reader can reproduce the calculation and distinguish the number of updates from the number of entries.
02
Apply the cap at each update
For the capped version of this fictional model, calculate two times the current state plus 1, then take the smaller of that result and 10. The states are 1, 3, 7, 10 and 10. At the third update, the raw result would be 15, so the cap changes it to 10. The next raw result is 21 and is again reduced to 10. This version therefore remains at 10 after reaching the cap. The cap is applied during every update, not merely to the final displayed number, which matters if later rules depend on the stored state.
03
Compare the paths without overinterpreting
Use columns for step, uncapped state, capped state and their difference. The differences from step zero through four are 0, 0, 0, 5 and 21. The paths first diverge at step three. Record the initial value, multiplier, added amount and cap above the table so another reader can reconstruct both sequences. These results describe the chosen recurrence only. A different model might oscillate, decrease or respond to outside inputs, so the exercise supports a specific comparison of rules rather than a general claim about how growing systems behave.