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It's #6.) in the following picture:

enter image description here

I know that this is a geometric series. Therefore, I planned on using the formula: $S_n$$ = \frac{t_1(r^n-1)}{r-1}$

Where:

$S_n$ = sum of the first $n$ terms

$r$ = common ratio

$n$ = the number of terms

$t_1$ = first term

I concluded that:

$S_n$ = 350

$n$ = 2200

$t_1$ = 20

And I need to use the formula and what I know to calculate $r$, aka the portion of the profit per cup.

Here's what I did:

$350$$ = \frac{20(r^{2200}- 1)}{r-1}$

$(r-1) 350$$ = 20(r^{2200}- 1)$

$\frac{350r-350}{20}$$ =r^{2200}-1$

$350r-17.5$$ =r^{2200}-1$

$350r-16.5$$ =r^{2200}$

$ =r^{2200}-350r+16.5$

Basically I was trying to isolate $r$, but I know what I ended up with looks wrong.

The answer in the book is $0.15 per cup.

I'm assuming that my approach might've been wrong? What exactly did I do wrong?

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Let $r$ be the amount collected per cup. The problem gave us

$$ 20 + 2200r = 350.$$

Solve this to get $r=0.15$.

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