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Originally I was looking for some simpler more intuitive way to appreciate the value of squaring something. I looked for how much it was increasing by every time I add $+1$ to the size of what was being squared. so in other words $3$ squared is $+5$ more than $2$ squared, and $4$ squared is $+7$ more than $3$ squared, and $5$ squared is $9$ more than $4$ squared. You can see that whenever you increase the number being squared by $1$, the amount increased itself increases by $+2$ more than the previous increase. ie, the increase itself increases by a rate of $+ 2$.

So

$$- 1^2 - 0^2 = +1$$ $$- 2^2 - 1^2 = +3$$ $$- 3^2 - 2^2 = +5$$ $$- 4^2 - 3^2 = +7$$ $$- 5^2 - 4^2 = +9$$

and

  • the difference between the $+3$ and $+1$ is $2$
  • the difference between the $+5$ and $+3$ is $2$
  • the difference between the $+7$ and $+5$ is $2$
  • the difference between the $+9$ and $+7$ is $2$ etc...

So I say, it's "increase bonus" is at a rate of +2. That's definitely not a correct mathematical term, but in my ignorance that's what i'm calling it.

This struck me as very interesting, because I used to make little wc3 mods as a kid and they gave tools that made same pattern. What I became interested in, was what if I wanted it to increase by a pattern of say +1, +2, +3, +4, +5. (for an increase bonus of +1). Or what if I wanted it to be +1, +4, +7, +10, +13 (an increase bonus of +3)?

My main question is, is there a formula where I can easily change the "increase bonus" to whatever rate I want? While I can do this manually, I don't know how to represent it as a simple, calculable formula.

My side question is... what is this formula / branch of mathematics actually officially called?

Mathemagician314
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Joshua Quesnel
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1 Answers1

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A polynomial of degree $n$ will have the $n$th finite difference equal to $n!$ times the leading coefficient of the polynomial. In your example, you have \begin{array}{|c|c|c|c|} \hline x & x^2 & \text{1st difference} & \text{2nd difference} \\ \hline 1 & 1 & & \\ 2 & 4 & 3 & \\ 3 & 9 & 5 & 2 \\ 4 & 16 & 7 & 2 \\ 5 & 25 & 9 & 2 \\ \hline \end{array}

If you want the bonus to be $+3$ instead, you can use $\frac32x^2$ instead of $x^2$: \begin{array}{|c|c|c|c|} \hline x & \frac32x^2 & \text{1st difference} & \text{2nd difference} \\ \hline 1 & \frac32 & & \\ 2 & 6 & \frac92 & \\ 3 & \frac{27}2 & \frac{15}2 & 3 \\ 4 & 24 & \frac{21}2 & 3 \\ 5 & \frac{75}2 & \frac{27}2 & 3 \\ \hline \end{array}

Or, if you want instead the third finite difference to be $+3$, you can use $\frac12x^3:$

\begin{array}{|c|c|c|c|c|} \hline x & \frac12x^3 & \text{1st difference} & \text{2nd difference} & \text{3rd difference}\\ \hline 1 & \frac12 & & & \\ 2 & 4 & \frac72 & & \\ 3 & \frac{27}2 & \frac{19}2 & 6 & \\ 4 & 32 & \frac{37}2 & 9 & 3\\ 5 & \frac{125}2 & \frac{61}2 & 12 & 3\\ \hline \end{array}

Andrew Chin
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