Lets say you had a person, who at 20 had his first child. He then had a child every 2 years, and ended up with 10 children. Everyone of his children followed the same pattern.
How old would he have to be to have 100 descendants? To have 1000 descendants?
He would be 38 when he finished having children.
He would be 40 when the first grandchild arrived.
He would be 42 when the next two arrived.
When he is 60 each of his children would have a child.
So at 60, he has his 10 plus his kids grandchildren (1+2+3+4+5+6+7+8+9+10) for a total of 65 descendents.
Now the oldest child is done having kids, so there are 9 more descendents when he is 62.
When he is 78, all his kids would be done having kids.
He would have 65 + 9 (when he was 62) + 8 (at 64) + 7 +6 +5 + 4 +3 +2 +1 = 110 descendents.
1+2+3+4 = 10. Therefore counting backwards, sometime in his 70th year he would have 100 descendents.
I’ll let someone else worry about formulating a formula or finding 1000.
Well, actually, when he is 90, his oldest would be 70 and would have his own 100 descendents. So when he is 110, all of his kids would have at least 100 descendents, so he would have at least 1000. But there is overlap between the generations so the year of his 1000 descendent would come much sooner.
This will happen when he’s 66 years old.
By that time, he’ll have 10 children, 85 grandchildren, and 20 grandchildren, for 115 total descendants. Two years prior he’d have 99 descendants (10, 79, 10).
I think I’ll let someone else take this one.
Whoops, that should be 20 great-grandchildren.
Okay, I’m bored, so I did the 1000 descendants question too. This happens when he is 98, when he will have 10 children, 100 grandchildren, 385 great-grandchildren, and 715 great-great-grandchildren, for a total of 1210 descendants. Two years prior he’d have 990 descendants (10, 100, 385, 495).