Compound interest formula $A= \frac{p((1+\frac rn)^{nt}-1)}{\frac rn} $

Question

I am doing an MBA assignment and I have given this formula

$$A=\frac{p((1+\frac rn)^{nt}-1)}{\frac rn} $$

Is this formula correct? I have usually seen $A=p(1+\frac rn)^{nt}$

The specific problem is

"You make monthly payments of $500 into a retirement annuity that earns 4% compounded monthly. What is the value of your account after 10 years? Round your answer to the nearest cent (two decimal places)."

Answer 1

Note that $p(1+\frac rn)^{nt}$ is the compounded value of a single payment. The value in 10 years is the sum of all monthly payments, which is

$$A=\sum_{k=0}^{nt-1} p \left(1+\frac rn \right)^{k}=\frac{p((1+\frac rn)^{nt}-1)}{\frac rn} $$