But is my math statement on the Plants vs. Zombies wiki mathematically correct?

Question

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$$2/28/23$$

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Back when I was more into researching mathematical analysis of video games, I came across this guide for modding PvZ$2$ which gave a link to this spreadsheet. So I made a copy (mainly because I wanted to see how plant leveling worked), and I'm unable to find how plant levels work, but I do see this statement when I hover over "Zombie Level" in the Settings sheet:

Stat Scaling is edited under ScaledProps in Zombie Properties.

$\color{white}{\text{.}}$

This can affect the HP, eatDPS and speed of a zombie.

$\color{white}{\text{.}}$

Stat multiplier formula:

$\operatorname{arg1}^{(\text{level}-1)}+\operatorname{arg2}\cdot(\text{level}-1)$

$\color{white}{\text{.}}$

Vanilla uses:

$\operatorname{arg1}=1.3$

$\operatorname{arg2}=0.05$

for all zombie's HP and eatDPS.

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Screenshot for reference

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Now here's the thing: I managed to find this statement as being correct and posted it on the PvZ$2$ wiki:

Fun fact: If you hack PvZ$2$ and set the level to $-4.16218$, the smallest amount of time that it would take something to recharge (calculated by min. base recharge time) would be approx. $4.2$ months (from $5$ seconds base recharge time) and the largest amount of time would be almost $6$ years (~$5.9$ years from $85$ seconds base recharge time)

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How my thought process probably went

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1. Take the formula.
2. Plug in $-4.16218$ for the level.
3. Take the reciprocal.
4. Multiply by the recharge time.
5. Take the reciprocal.
6. Convert what you got from seconds to whatever

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Now here's the thing. While this should help with calculating recharge time, it doesn't help with plant damage/plant health.

(I think I was ironically wanting just the recharge, and I say ironically because it's kind of ironic because I originally wanted to get the formula for calculating plant damage and plant health, not recharge time)

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$$5/23/23$$

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I went onto the PvZ$2$ Fandom and tried replicating what I had done, but I was unable to replicate the formula any way I could. I eventually decided to do$$a(1.3^{b-1}+0.05^{b-1})^{-1}$$where $a$ is the recharge time and $b$ is the level number, which I'm pretty sure it represents actual recharge times accurately, although not $100$%.

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This is the table of level to length of recharge time compared to what I said on how much I was off by:

Level	Length of recharge time (actual)	Length of recharge time (what I said)	How much I was off
$-4.16218$ ($5$ sec recharge time)	$18.06230212933333$ hours	$4.2$ months	$4.17525714749$ months
$-4.16218$ ($85$ sec recharge time)	$14.02049553196347$ years	$6$ years	$8.02049553196347$ years

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Therefore, my statement on the Plants vs. Zombies $2$ wiki is mathematically incorrect, and it is not only a lie, but a huge lie in that matter.

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My question

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Is my new statement correct, or what would be the correct statement in this case?

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Mistakes I might have made

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1. pretty much everything honestly, there's no actual way to calculate this type of stuff so I honestly don't know if I have done anything wrong or not.

Answer 1

Okay, so it turns out I am, in fact, wrong.

As @Jaap Scherphuis pointed out, I used $0.05^{b-1}$ instead of $0.05\times(b-1)$

In the first part you use $0.05\times(b-1)$, in the second part $0.05^{b-1}$. Which one is correct?

It gets worse from here. Using this formula to recalculate my data, it turns out that I was incorrect for both calculations, meaning that originally I was correct.

New table:

Level	Length of recharge time (actual)	Length of recharge time (what I said)	How much I was off
$-4.16218$ ($5$ sec recharge time)	$\require{cancel}\cancel{18.06230212933333\text{ hours}}$ $4.2$ months	$4.2$ months	$\cancel{4.17525714749\text{ months}}$
$-4.16218$ ($85$ sec recharge time)	$\cancel{14.02049553196347\text{ years}}$ $6$ years	$6$ years	$\cancel{8.02049553196347\text{ years}}$

$\small\small\small\small\small\text{I'm horrible when it comes to solving for unknown variables am I}$ :\