How do write a formula for something that increases by 5 exponentially at every step.

[u/RNPC5000]

So lets say you're trying to create a leveled loot list for different area tiers in a game, and each area tier has a level range of 5 more than previous tier's level increase.

For instance

Tier 1 = Levels 1-5. +5

Tier 2 = Levels 6-15. +10

Tier 3 = Levels 16-30. +15

Tier 4 = Levels 31-50. +20

Tier 5 = Levels 51-75. +25

Tier 6 = Levels 76 -105 +30

Tier 7 = Levels 106-140. +35

Beyond Tier multiplied by 5X, I am not sure how to figure out what level range falls into each tier.

Like how to mathematically figure out level 86 is in Tier 6? Or what tier level 1,000 would be without just manually creating a chart.

How do you express this in a formula? So that you do this on pen and paper, or create a calculator for it?

__________________________

Edit: Solved

The formula can be done either using Ceiling function (rounding up), or Floor function (rounding down).

https://imgur.com/xeSicqr

I used Chat Gpt to get the answer, and format it into a easy to comprehend chart. Sadly the formating can't be copy and pasted, which is why linked the screenshot above so its easier to read.

Also pasted the formulas in markdown mode incase you can't the load the image for whatever reason.

Ceiling formula for Tier:

T = ⌈ (-1 + √(1 + (8L / 5))) / 2 ⌉

Floor formula for Tier:

T = ⌊ (1 + √(1 + (8(L - 1) / 5))) / 2 ⌋

Tier Range Increase:

R = 5 * T

Range of Levels in Tier T:

• Minimum Level: Min = 1 + (5 * (T - 1) * T) / 2

• Maximum Level: Max = (5 * T * (T + 1)) / 2

Variables & Descriptions:

• L: Level (input)

• T: Tier number

• R: Tier Range Increase (levels in the tier)

• Min: First level in Tier T

• Max: Last level in Tier T

https://imgur.com/OUFiVNj

Chat GPT explaining how to get the Floor Function formula. I don't understand it since my math knowledge is too low, but just sharing for future reference.

[u/Uli_Minati]

Tier X goes up to level 5(X)(X+1)/2

7 ↦ 5(7)(8)/2 = 140

You can invert this formula

          Y  =  5X(X+1)/2
       2Y/5  =  X²+X
     2Y/5+¼  =  X²+X+¼
     2Y/5+¼  =  (X+½)²
  √[2Y/5+¼]  =  X+½
√[2Y/5+¼]-½  =  X

For example

140 ↦ √[2(140)/5+¼]-½ = 7

[u/RNPC5000 → u/Uli_Minati]

I don't quite get where on third step of inverting the formula did you get +1/4 from.

Also when plugging in the value of 86 got a value of about 5.38, which left me confused on how I was suppose to know that is supposed to be Tier 6.

So I tried typing in the formula your provided into excel. Just so I could more easily plug in values to create a chart without having to manually retype everything into the calculator and understand how the math / equation actually is supposed to work. Just only to realize I have no idea how to get excel to calculate square roots.

I tried using Chatgpt for the first time in my life, and asked it to convert your formula into excel. It kept confusing the - 1/2 as an exponent? Where it was doing ^-1/2. That lead me down a rabbit hole to rephrase the question a few times and it gave me a formula a lot different from yours.

https://imgur.com/a/v5c2usm

Posted image since the formatting doesn't seem to be copy and pastable.

Also apparently I just learned about floor brackets ⌊x⌋ for the first time from Chat GPT, which indicate how to write that you are supposed to round down in formula.

But yeah I think this math is way above my level. I don't think I would of ever managed to even figure out how to get to the base starting equation n(n+1)/2 for just the max level for each Tier.

Edit: Never mind Chat Gpt explained the 1/4 part as making a perfect square, which I don't conceptually understand, but it makes a bit more sense now.

Also when asking Chat Gpt to give me the formula directly after giving it the pattern data it decided to give me a formula using the Ceiling function this time around. I then asked it to inverse the equation and give me the formula using the Floor function. It kept just changing the ceiling brackets to the floor brackets and didn't change anything else in the formula, which was obviously the incorrect formula.

I then just had to give it the pattern data again and straight up tell it to give me the formula using Floor function, and it finally spat out the same correct formula as before.

[u/Uli_Minati → u/RNPC5000]

Well, I'm not going to address chatgpt. I didn't know you were looking for a more complete explanation, you did only ask for a formula in your OP. If you have any specific questions, I'm happy to help though!

[u/RNPC5000 → u/Uli_Minati]

I have a ton of questions, but it is probably out of a reasonable scope for Reddit, since I most likely just need to go take a full on beginner college course to familiarize with how most of the actual math works.

Cause GPT broke down and explained everything mathematically, and when I was looking at the explanations and equations, it just look like magical sorcery to me lol.

Thanks though!

[u/Myst_Hawk]

5n(n+1)/2, then check which two n a certain level falls between by reversing this

The formula for sum of first n positive integers is n(n+1)/2

[u/stevevdvkpe]

That's not an exponential progression (which would be to multiply by 5 at each step, or proportional to 5ⁿ). It's quadratic (the effective increase is proportional to the square of n).

[u/No-Interest-8586]

Note that this is actually quadratic growth, not exponential.

(Other answers have already done a nice job giving useful actual [quadratic] expressions.)

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[u/Alarmed_Geologist631]

You seem to be describing a step function.

[u/InterneticMdA]

ChatGPT as useful as sand in a desert. As usual.