a$.$ Suppose that between the ages of $22$ and $30,$ you contribute $$1000$ per year to a $401(k)$ and your employer contributes $$500$ per year on your behalf. The interest rate is $9\mathrm{.}4\%$ compounded annually. What is the value of the $401(k)$ after $8$ years? b$.$ Suppose that after $8$ years of working for this firm, you move on to a new job. However, you keep your accumulated retirement funds in the $401($k$).$ How much money will you have in the plan when you reach age $65?$ c$.$ What is the difference between the amount of money you will have accumulated in the $401(k)$ and the amount you contributed to the plan?

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a$.$ The value of the $401(k)$ after $8$ years is $

$($Do not round until the final answer. Then round to the nearest dollar as needed.$)$

Formulas

In the following formulas, $P$ is the deposit made at the end of each compounding period, $r$ is the annual interest rate of the annuity in decimal form, $n$ is the number of compounding periods per year, and $A$ is the value of the annuity after $t$ years.

$A=\frac{P[(1+r{)}^t-1]}{r},A=\frac{P[(1+\frac{r}{n}{)}^{nt}-1]}{(\frac{r}{n})}P=\frac{A(\frac{r}{n})}{[(1+\frac{r}{n}{)}^{nt}-1]}$

In the following formulas, $P$ is the principal amount deposited into an account, $r$ is the annual interest rate in decimal form, $n$ is the number of compounding periods per year, and $A$ is the future value of the account after $t$ years.

$A=P(1+r{)}^t,A=P(1+\frac{r}{n}{)}^{nt}$