Word Problem for Number of Seeds Required with Diminishing Returns

Question

So I've been trying to wrap my brain around this but it eludes me. The word problem would look something like this:

What is the minimum amount of seeds I need to buy to plant 850 seeds when only 60% of these planted seeds will produce a seed of their own. I can continue planting over and over until I reach my last seed.

I am looking for a formula I can use and plug into Excel - but a mathematical equation would be just great!

Thanks

Answer 1

You are just summing a geometric series. If you start with $n$ the next generation gives you $0.6n$, the one after that gives $0.6^2n$, then $0.6^3n$ so you want $$850=n(1+0.6+0.6^2+0.6^3+\ldots )=\frac n{1-0.6}\cdot 850\\n=0.4\cdot 850=340 $$

Answer 2

1 seed grows 2.5 seeds on average, so divide 850 by 2.5 and you have your answer. You can use infinite geometric series formula to find expected number of seeds for 1 seed.