Impossible Interest question.

Hello, skg94!

Your intentions were honorable.
You must have made some elementary errors.

Quote:

At the end of each quarter-year, Aaron makes a $625 payment into a mutual fund
that earns an annual percentage rate of 6%, compounded quarterly.
The future value, of aaron's investment is: . $FV \:=\: R\frac{(1+i)^n -1}{i}$
. . where $n$ = number of equal periodic payments of $R$ dollars,
. . and $i$ = interest rate per period.
After how long will aaron's investment be worth 1 million dollars?

We have: . $FV \,=\, 1,\!000,\!000,\;\;R \,=\,625,\;\; i \,=\, \tfrac{6\%}{4} \,=\, 0.015$

Substitute: . $625\,\frac{1.015^n - 1}{0.015} \:=\:1,\!000,\!000$

. . . . . . . . . $1.015^n - 1 \:=\:24 \quad\Rightarrow\quad 1.015^n \:=\:25$

Take logs: . $\ln(1.015^n) \:=\:\ln(25) \quad\Rightarrow\quad n\!\cdot\!\ln(1.015) \:=\:\ln(25)$

Hence: . $n \;=\;\frac{\ln(25)}{\ln(1.015)} \;=\; 216.1971659$

$\text{Therefore, it will take }217\text{ quarters }=\: 54\tfrac{1}{4}\text{ years.}$ .**

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** . $\text{If }n = 216\text{, he will have "only" }\$996,946.64$