You plan to invest a part of your paycheck(a fixed monthly investment) to finance your children's higher education. You want to have $\$1,000,000$ accumulated in the account at the end of $20$ years from now. If the account pays $0.5\%$ interest each month, how much should you deposit each month during next $20$ years to accomplish this financial objective?
Let $x_n$ be the money after $n$ months and $x_0=0$
$p$ be money deposit each month
$x_{n+1}=(x_n+p)0.005+x_n+p=1.005x_n+1.005p$
by solving this I obtained $x_n=(201p)(1.005)^n-201p$
After $20$ years, $$1000000=(201p)(1.005)^{240}-201p$$
then $p=2153.54$
Is my attempt correct?