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I'm learning precalculus via an open source book and it states something I'm confused about. Here is the quote:

Suppose you have an account with annual interest rate r which is compounded n times per year. We let i = $\frac{r}{n}$ denote the interest rate per period. Suppose we wish to make ongoing deposits of P dollars at the end of each compounding period. Let $A_k$ denote the amount in the account after k compounding periods. Then $A_1$ = P, because we have made our first deposit at the end of the first compounding period and no interest has been earned. During the second compounding period, we earn interest on $A_1$ so that our initial investment has grown to $A_1(1 + i)$ = P(1 + i) in accordance with Equation 6.1. When we add our second payment at the end of the second period, we get $A_2 = A_1(1 + i) + P = P(1 + i) + P = P(1 + i)(1 + \frac{1}{1+i})$.

It then goes on showing iterations of $A_3, A_4...$

Now, I get that at $A_2$ we have $A_1(1 + i) + P$. And I get this is equal to P(1 + i) + P even more so because P(1 + i) is our initial principal plus one compounding period of interest and the next + P is the additional principal we added this time around.
What I don't get is how does P(1 + i) + P = P(1 + i)(1 + $\frac{1}{1+i})$?

If I try plugging numbers into this it doesn't add up. Lets say P = 10 and lets say i = 2%

then P(1 + i) + P = 10(1 + .2) + 10 = 20.20.

Now lets plug the same values into P(1 + i)(1 + $\frac{1}{1+i})$ =

10(1 + .2)(1 + $\frac{1}{1+.2})$ =

10.20(1 + $\frac{1}{1+.2})$ =

(10.20 + $\frac{10.20}{1.2})$ =

10.20 + 8.5 = 18.70.

And since these don't add up, I don't see how P(1 + i) + P = P(1 + i)(1 + $\frac{1}{1+i})$? Am I doing wrong here?

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    $\begingroup$ To begin with $10(1.2)=12$, not $10.2$ $\endgroup$ – saulspatz Jan 22 '20 at 22:40
  • $\begingroup$ to add to the list 2% is 0.02 not 0.2 $\endgroup$ – user645636 Jan 22 '20 at 22:44
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The expression you are querying has simply used $$(1+i) \frac{1}{1+i} = 1$$

The reason your numerical example fails is because you've got a bit confused over the decimal representation of 2% which is 0.02 not 0.2. This means near the bottom where you have 8.5 you should have 10.

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