Suppose you invest $$ \$ 130 $$ a month for 4 years into an account earning $$ 8 \% $$ compounded monthly. After 4 years, you leave the money, without making additional deposits, in the account for another 29 years. How much will you have in the end?

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UpStudy AI · Accepted Answer

**Step 1. Compute the monthly interest rate**

The stated annual interest rate is $$ 8\% $$, and because it is compounded monthly, the monthly interest rate is

$$
r = \frac{0.08}{12} \approx 0.0066667.
$$

---

**Step 2. Compute the future value at the end of 4 years with monthly deposits**

You make monthly deposits of $$ \$130 $$ for 4 years. The number of deposits is

$$
n = 12 \times 4 = 48.
$$

For an annuity with regular deposits, the future value after $$ n $$ deposits is given by

$$
FV_4 = 130 \times \frac{(1 + r)^n - 1}{r}.
$$

Substitute $$ r = 0.0066667 $$ and $$ n = 48 $$:

$$
FV_4 = 130 \times \frac{(1 + 0.0066667)^{48} - 1}{0.0066667}.
$$

Calculating the term $$ (1 + 0.0066667)^{48} $$:

$$
(1.0066667)^{48} \approx 1.3746.
$$

Thus,

$$
FV_4 \approx 130 \times \frac{1.3746 - 1}{0.0066667} \approx 130 \times \frac{0.3746}{0.0066667} \approx 130 \times 56.19 \approx 7305.
$$

So, after 4 years, the account balance is approximately $$ \$7305 $$.

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**Step 3. Grow the 4‐year amount for an additional 29 years**

After 4 years, no further deposits are made and the accumulated amount is left to compound for another 29 years. Since the compounding is still monthly, the total number of compounding periods for the 29 years is

$$
m = 29 \times 12 = 348.
$$

The final future value after the additional 29 years is

$$
FV_{\text{final}} = FV_4 \times (1 + r)^m.
$$

Substitute the values:

$$
FV_{\text{final}} \approx 7305 \times (1.0066667)^{348}.
$$

Calculate $$ (1.0066667)^{348} $$:

$$
(1.0066667)^{348} \approx 10.01.
$$

Thus,

$$
FV_{\text{final}} \approx 7305 \times 10.01 \approx 73,\!365.
$$

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**Final Answer**

The final amount in the account, after 4 years of deposits and an additional 29 years of compounding, is approximately $$ \$73,\!365 $$.
After 4 years of monthly deposits and 29 years of compounding, the account will have approximately \$73,365.

Suppose you invest a month for 4 years into an account earning compounded
monthly. After 4 years, you leave the money, without making additional deposits, in the
account for another 29 years. How much will you have in the end?

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Suppose you invest a month for 4 years into an account earning compounded monthly. After 4 years, you leave the money, without making additional deposits, in the account for another 29 years. How much will you have in the end?