I have a hypothetical population growth question, and I can't solve it.

[MathGuruNot]

Hi, lets say you have a couple and they have 2.5 children (I know this is biologically impossible, I'm just looking for how the numbers are multiplied).

The children have 2.5 children.

This goes on for 100 or 101 generations (let's solve both).

But people only live for 43 years (this isn't important mathematically, but it helps make sense for the final result).

So finally, how many people were born in that final 100th generation?

What about if it was a 101st generation?

If the children and their parents were both alive (no grandparents as 43 is when people expire), how many people would be alive during the 100th generation? What about the 101st generation?

Sorry I have to ask. I wish my math was better.

2.5^100 - 2.5^99 for a final generation is impossibly large, because the exponential growth isn't taking into account death at each generation.

This is based on a hypothetical scenario I read about and I wanted to know if their approximate answer was correct.

Thank you for your time and patience.

 

[Dr.Peterson]

There are a few important parameters you haven't identified; for one thing, how old would the parents be when each child was born?

What's most important to me here is that a "generation" is not as well-defined as you think. Since the children in one "generation" are not all born at the same time, generations would blur together, so that if, say, children were born when their parents are 25 and 30, the spread from oldest to youngest in each generation would grow to about 500 years at the 100th generation! It would be impossible even to define what "during the 100th generation" means. (I might be the 13th generation from some ancestors 400 years ago, but the 16th generation from others.)

But since you want to ignore reality and suppose that 2.5 children, whatever that means, appear at once, let's go for that. Your calculation misses something extremely important, and it is not death. It's that it takes 2 people to produce the 2.5, so at each generation, the number in the next generation is multiplied not by 2.5 but by 2.5/2 = 1.25. So the number in generation #100 is 2*1.25^99 (the first generation having been 2*1.25^0). This is still, of course, a big number, though not nearly as big as yours: about 8 billion. Then, if you suppose that at whatever time you are asking about the parents are also alive, you's just add 2*1.25^98 for that previous generation, for a total of about 14 billion.

 

[MathGuruNot]

Thank you. I wasn't looking at the specifics, just an exponential hypothetical where each generation is 2.5x the previous generation. Obviously all those other factors mean something if I'm preparing some sort of paper on theoretical population growths, but I wasn't.

 

[HallsofIvy]

If the initial number is A, so f(0)=A, then f(1)=2.5 times A=2.5A, f(2)=2.5(2.5)A=2.5^2A, f(3)=2.5(2.5^2A)=2.5^3A, f(4)= 2.5(2.5^3A)=2.5^4A. So what is your guess for f(n)?