Present value and discount function

Question

What deposit made today will provide for a payment of $ \$1000$ in $1$ year and $ \$2000$ in $3$ years, if the effective rate of interest is $0.075$. Answer is $\$2540.15$

I have calculated $d=0.069$ by using $i=\frac{d}{1-d}$

Then by using $A(3)=A(0)(1-d)^{-3}$, I don't get the answer

Answer 1

Hint: if $X$ is the deposit, and after one year you have a payement of $1000$, the residual value after this year is $$ X_1=X(1+i)-1000 $$ that after two years becomes: $$ \left(X(1+i)-1000\right)(1+i)^2=2000 $$ solve for $X$ and you have the result.

Answer 2

Let $A_0$ denote the initial amount, and $A_i$ the amount amount $i$ years. We have $$A_1=A_0(1+i)-1000\;\;\;\;\;\;A_2=3000=A_1(1+i)^2$$ Substituting for $A_1$ we see that $$3000=(A_0(1+i)-1000)(1+i)^2=A_0(1+i)^3-1000(1+i)^2\implies A_0=\frac {3000+1000(1+i)^2}{(1+i)^3}$$ Plugging in $i=.075$ yields $$A_0\sim\$3345.1143$$

Note: I see that the problem was revised after I posted. It is, of course, easy to change $\$3000$ to $\$2000$ in the above.