Suppose you invest $$ \$ 190 $$ a month for 8 years into an account earning $$ 9 \% $$ compounded monthly. After 8 years, you leave the money, without making additional deposits, in the account for another 28 years. How much will you have in the end? $$ \$ $$ Suppose instead you didn't invest anything for the first 8 years, then deposited $$ \$ 190 $$ a month for 28 years into an account earning $$ 9 \% $$ compounded monthly. How much will you have in the end? $$ \$ $$

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We want to compute the future value of monthly deposits of $$ \$190 $$ into an account earning $$ 9\% $$ per year compounded monthly. In all cases the monthly interest rate is

$$
r=\frac{0.09}{12}=0.0075.
$$

A useful formula for the future value of an annuity (regular deposits) made at the end of each period is

$$
\text{FV} = PMT\cdot\frac{(1+r)^n-1}{r},
$$

where
- $$ PMT $$ is the payment per period,
- $$ n $$ is the total number of deposits,
- $$ r $$ is the interest rate per period.

Below we solve each part step by step.

---

**Part 1.**  
You deposit $$ \$190 $$ per month for 8 years (that is, $$ 8\times12=96 $$ months) and then leave the money to compound for an additional 28 years (i.e. $$ 28\times12=336 $$ months) with no additional deposits.

1. **Accumulation after 8 years:**  
   Compute the future value after the deposit period using the annuity formula:

$$
   \text{FV}_8 = 190\cdot\frac{(1.0075)^{96}-1}{0.0075}.
   $$

2. **Growth during the following 28 years:**  
   After 8 years, no further deposits are made. The balance simply grows with compound interest for 336 months. Thus, multiply the amount above by $$(1.0075)^{336}$$:

$$
   \text{Total FV} = \text{FV}_8\cdot (1.0075)^{336} = 190\cdot\frac{(1.0075)^{96}-1}{0.0075}\cdot (1.0075)^{336}.
   $$

Using an approximation for the exponents:  
- $$ (1.0075)^{96}\approx2.0485 $$ (since $$ \ln(1.0075)\approx0.007472 $$ and $$ 96\times0.007472\approx0.7173 $$, so $$ e^{0.7173}\approx2.0485 $$),  
- $$ (1.0075)^{336}\approx12.30 $$ (since $$ 336\times0.007472\approx2.509 $$ and $$ e^{2.509}\approx12.30 $$),

we get

$$
\text{FV}_8 \approx 190\cdot\frac{2.0485-1}{0.0075} 
= 190\cdot\frac{1.0485}{0.0075} 
\approx 190\cdot139.80 \approx 26,\!562.
$$

Then, after 28 more years the final value is approximately

$$
\text{Total FV} \approx 26,\!562\times12.30 \approx 326,\!700.
$$

---

**Part 2.**  
Now suppose you do not deposit during the first 8 years. Instead, you begin depositing $$ \$190 $$ per month only for 28 years (336 months) into the same account.

The future value is given directly by the annuity formula with $$ n = 336 $$:

$$
\text{FV} = 190\cdot\frac{(1.0075)^{336}-1}{0.0075}.
$$

Using our approximation $$ (1.0075)^{336}\approx12.30 $$, we have

$$
\text{FV} \approx 190\cdot\frac{12.30-1}{0.0075}
= 190\cdot\frac{11.30}{0.0075}.
$$

Calculate the inner fraction:

$$
\frac{11.30}{0.0075}\approx1506.67,
$$

and then

$$
\text{FV}\approx 190\times1506.67\approx 286,\!267.
$$

---

**Final Answers:**

1. After 8 years of deposits followed by 28 years of growth, the amount is approximately $$\$326,\!700$$.

2. If you begin depositing only in the later 28 years, the account value at the end is approximately $$\$286,\!267$$.
After 8 years of monthly deposits of \$190 at 9% interest compounded monthly, the account grows to approximately \$326,700 over the next 28 years. If you instead start depositing \$190 monthly only during the 28 years, the account will have about \$286,267 at the end.

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