Hi Ava,

Good question. Try breaking down the values week by week, starting with a small amount:

Week 0: Bank ($0), Income ($0), Raise ($0), Raise Raise ($1) Week 1: Bank ($0), Income ($0), Raise ($1), Raise Raise ($1) Week 2: Bank ($0), Income ($1), Raise ($2), Raise Raise ($1) Week 3: Bank ($1), Income ($3), Raise ($3), RR ($1) Week 4: Bank ($4), Income ($6), Raise ($4), RR ($1) Week 5: Bank ($10), Income ($10), Raise ($5), RR ($1) Week 6: Bank ($20), Income ($15), Raise ($6), RR ($1) Week 7: Bank ($35), Income ($21), Raise ($7), RR ($1) Week 8: Bank ($56), Income ($30), Raise ($8), RR ($1)

Here, the raise is growing the same amount each week ($1, $2, $3, etc.)
The income is growing to the 2nd power, and is roughly: 1/2 x^2.

The bank balance is growing to the 3rd power, and is roughly: 1/6 x^3

See https://instacalc.com/52176

-------------------------------------------------------------------------------------------------------------------------- Can we keep going "down" (taking derivatives) beyond the raise?

Well, why not? If the raise is $100/week, if we take the difference again we see it drops to 0 (there is no "raise raise", aka the raise is always steady). But, we can imagine the case where the raise itself is raising (week1 raise = 100, week2 raise = 200). Using our intuition: if the "raise raise" is constant, the raise is linear (something * n), the income is quadratic (something * n2) and the bank account is cubic (something * n3). And yes, it's true!
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But how does this work? I've been trying for hours but I don't understand. The only way I could think of for the quadratic one was:

(100 * n^2) - C = income.

But with this I'm not even sure which 100 I'm using (raise?).

Then there's the one with the bankaccount and ^3 and I'm just completely lost. I've tried everything. Please explain!

Edit: Also, what is the "something" in something * n/^2/^3

Thanks Michael -- yep, many textbook give you the how but not the why. Computers can crunch the equations, but we want to learn to read and interpret them ourselves. Hope you enjoy the site!