TITLE: Is it possible to find a ratio/rate for this sequence?
QUESTION [0 upvotes]: n, n + n^2, n + n^2 + n^3, ...
OR...
n, n(n+1), n(1 + n + n^2), ...
OR
3, 3+9, 3+9+27, 3+9+27+81
Not related to academics. I was trying to figure out a ratio/rate so I can model it into an understandable curve for work, but could not figure out how to do it for the love of my life. (Work is not math related at all)
The concept is this: Generation 1 worker produces 1 good but can train 3 other workers (Generation 2). Generation 2 workers can now each produce 1 good and also train 3 other workers (Generation 3), meanwhile Generation 1 worker is STILL producing his 1 good but reached his limit in training any more workers. Generation 3 workers likewise produce 1 good and can each train 3 others; Generation 2 is still producing 1 good each but cannot train anymore; Generation 1 is still producing 1 good but cannot train anymore.
I hope you get the idea. Please please help! I am desperate!

REPLY [0 votes]: $n + n^2 + n^3 + \cdots + n^k = \dfrac{n^{k+1}-1}{n-1}-1$
e.g.
$3+9=3^1+3^2=\dfrac{3^3-1}{2}-1 = 12$
$3+9+27=3^1+3^2+3^3=\dfrac{3^4-1}{2}-1 = 39$
$3+9+27+81=3^1+3^2+3^3+3^4=\dfrac{3^5-1}{2}-1 = 120$
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