Compute successive differences over levels for perfect cubes until all the differences at a level are the same. What do you notice?
Perfect cubes: 1, 8, 27, 64, 125, 216
Level 1 differences: 8 - 1 = 7, 27 - 8 = 19, 64 - 27 = 37, 125 - 64 = 61, 216 - 125 = 91 7, 19, 37, 61, 91, ...
Level 2 differences: 19 - 7 = 12, 37 - 19 = 18, 61 - 37 = 24, 91 - 61 = 30 12, 18, 24, 30, ...
Level 3 differences: 18 - 12 = 6, 24 - 18 = 6, 30 - 24 = 6 6, 6, 6, ...
Observation: At level 3, all differences are the same (6). This shows that for cubes, the third level of successive differences is constant at 6.
Explanation
The chapter compares successive differences for squares and cubes. For squares, the second level differences are constant (all 2). For cubes, the third level differences are constant (all 6). This pattern continues for higher powers.
Solution Steps
Step 1: Perfect cubes: 1, 8, 27, 64, 125, 216
Step 2: Level 1 differences: 8-1=7, 27-8=19, 64-27=37, 125-64=61, 216-125=91
Step 3: Level 2 differences: 19-7=12, 37-19=18, 61-37=24, 91-61=30
Step 4: Level 3 differences: 18-12=6, 24-18=6, 30-24=6
Step 5: All level 3 differences equal 6