Vtyjkyuk

If I invest $$2000$/yr in a tax sheltered annuity at $7%$, where $A_n$ is the amount at $n$ years... what is the recurrence relation? I know my initial condition $A_0$ is $2000$. And for some reason the answer I got is $A_n = (1.12)^2 cdot A_0$.

Any help would be appreciated, I’m not sure where I went wrong!

At the end of the $n$th year, you invest $$2000$ and earn $7%$ interest on what you earned the previous year. Hence,
$$A_n = $2000 + 0.07A_n - 1, n geq 1$$
which is the recurrence relation. The initial value in your account is $A_0 = $2000$.

Let's examine what happens during the first few years.
beginalign*
A_0 & = $2000\
A_1 & = $2000 + 0.07A_0\
& = $2000 + 0.07 cdot $2000\
& = $2000(1 + 0.07)\
A_2 & = $2000 + 0.07A_1\
& = $2000 + 0.07[$2000(1 + 0.07)]\
& = $2000 + 0.07 cdot $2000 + 0.07^2 cdot $2000\
& = $2000(1 + 0.07 + 0.07^2)\
A_3 & = $2000 + 0.07A_2\
& = $2000 + 0.07[$2000(1 + 0.07 + 0.07^2)]\
& = $2000 + 0.07 cdot $2000 + 0.07^2 cdot $2000 + 0.07^3 cdot $2000\
& = $2000(1 + 0.07 + 0.07^2 + 0.07^3)
endalign*
which suggests that
$$A_n = $2000(1 + 0.07 + 0.07^2 + 0.07^3 + cdots + 0.07^n) = $2000sum_k = 0^n 0.07^k$$
which you can prove by induction.

As for finding an explicit formula, you can use the geometric series formula
$$A_n = a(1 + r + r^2 + cdots + r^n) = afrac1 - r^n + 11 - r, r neq 1$$
with $a = $2000$ and $r = 0.07$.