Eli5 Wtf is the difference between geometric growth and arithmetic growth
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An arithmetic series is one where you add. A geometric series is one where you multiply. For example, if we start at 1 and add 2, we get 1,3,5,7,9,…, an arithmetic series (the odd numbers). If we start at 1 and *multiply* by 2, we get 1,2,4,8,16,…, a geometric series.
We can also talk about the arithmetic average of two numbers which is (x+y)/2, and the geometric average which is √(x×y).
**Arithmetic**
Like others have said, arithmetic means you add. If you had a bank account and you put in 10 bucks each week and charted how much money you had in the bank each week, that would be arithmetic growth.
That formula would look like: (Money in bank) = 10 bucks x (the number of weeks)
It can be a little confusing because of the “times” in there when I just said it was adding, but that’s just a faster way of writing it.
Another way that show the adding would be: (Money in bank at week three) = ($10 from week one) + ($10 from week two) + ($10 from week three)
Add more weeks and you’d just add more ten dollars. If you wait one week you have $10. If you wait 10 weeks you have $100. If you wait a full year, you have $520.
**Geometric**
This is a bit harder to give an example for but it’s basically growth by multiplying instead of adding. It tends to start off growing slow and then gets much faster as time goes on.
Imagine you found a genie and wished to be rich, but this genie is a dick mathematician. He says sure, but how rich you get is going to depend on how patient you are. He gives you a magic bank account that you can put in your name whenever you want but once you do it stops gaining money, so you need to make sure you wait until it’s worthwhile.
The way it works is that for how every many weeks it’s been, it worth that many weeks **times** that many weeks.
The formula for that would be: (Money in bank) = (number of weeks)^2 .
Again, the exponent makes it look weird but that’s just a faster way of writing it.
Keeping in the “multiply” it would be: (Money in bank) = (number of weeks) x (number of weeks).
This is where the difference between the two really shows. Week one you would have $1 (much lower than the first example). Week ten you would have $100 (now equal to the first example at 10 week). At a year it would be worth $2704.
Arithmetic growth: You start at some point (say 100) and then add 20 every day. So 100, 120, 140, 160, 180, 200… This is what happens if you have a jar of money and put a $20 bill in it every day.
Geometric growth: You multiply by 20%, and add that much, every day. So (with some rounding) 100, 120, 144, 172.8, 207.36, 248.832… This is what happens if you have a patch of bacteria in a lab dish where they get plenty of food.
You see, bacteria make baby bacteria, and then within hours those baby bacteria grow up and make more baby bacteria, so not only are you adding 20% every day, but the 20% you added yesterday is going to add even more today. So tomorrow’s count will be a bigger increase from today’s count, than today’s count was from yesterday’s count.
Geometric growth can add up *fast*. If you continue arithmetic growth for 90 days, you’ll have $1900 in the jar. If you continue growing the bacteria from the same 100 starting point, after 90 days you’ll have 1,337,556,524.
(Real world applications of geometric growth are often simplified cartoon stories that don’t take into account the real-world usually has limiting factors. If you calculate this bacteria geometric sequence for 2 years, it will predict you’ll have more bacteria than atoms on Earth, which is rather ridiculous. That’s because using a geometric sequence doesn’t account for when the bacteria reach the end of the dish, run out of food, choke themselves on waste products, and so on.)
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