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. \begin{align*} 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) \end{align*} which suggests that $$A_n = \$2000(1 + 0.07 + 0.07^2 + 0.07^3 + \cdots + 0.07^n) = \$2000\sum_{k = 0}^{n} 0.07^k$$ which you can prove by induction.
As for finding an explicit formula, you can use the geometric series formula $$A_n = a(1 + r + r^2 + \cdots + r^n) = a\frac{1 - r^{n + 1}}{1 - r}, r \neq 1$$ with $a = \$2000$ and $r = 0.07$.