Why would you assume 3 kids for every generation when birth rates are currently below replacement level? Also, add in rich people marrying people of similar socioenomic status, and you end up with quite wealthy families down the line.

For the wealthy when people refer to growth versus consumption the context is almost always against the return generated from their portfolio, and the two are separate figures. That "4% growth above inflation" does not include what the capital owner is drawing for his own income. It's almost certainly more like an 8% - 10% return gives 3% to taxes and inflationary pressure, 3% - 4% for income, and the rest to growth.

And, yes, it's quite easy for the wealthy to get investments that return 7% - 8% with low to modest risk of loss. It's much harder to get more than that, but it's not difficult to generate substantial returns from even a modest portfolio in the low 7-figure range.

5% sorry. It's a reasonable long term return.

25 years per generation.

Year 0: 10m Year 25: 33.8m. 3 kids, give them 10m each, leaving you with 3.8m for the rest of time

This repeats as long as you want.

Over 100 years at 5% you have 1.315 billion over 120 people, which is 10.95m each.

At 4% you'd have to have a 29 year generation, which is certainly more realistic (I know very few people who had the first kid under the age of 30). And would everyone have 3 kids? Don't forget to account for kids who die before they reproduce too.