\begin{tabular}{l} Acme Annuities recently offered an annuity that pays $$ 6.9 \% $$ compounded monthly. What equal monthly deposit should \ be made into this annuity in order to have $$ \$ 172,000 $$ in 11 years? \ The amount of each deposit should be $$ \$ \square $$ \ (Round to the nearest cent.) \ \hline\end{tabular}

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We are given an annuity with a nominal rate of $$ 6.9\% $$ compounded monthly. The deposit period is 11 years with deposits made monthly, so there are

$$
n = 11 \times 12 = 132 \text{ deposits}.
$$

The periodic (monthly) interest rate is

$$
i = \frac{6.9\%}{12} = \frac{0.069}{12} \approx 0.00575.
$$

For an annuity in which equal deposits $$ D $$ are made at the end of each month, the future value (FV) is given by

$$
\text{FV} = D \cdot \frac{(1+i)^n - 1}{i}.
$$

We are told that

$$
\text{FV} = \$172{,}000.
$$

We solve for $$ D $$ by rearranging the formula:

$$
D = \text{FV} \cdot \frac{i}{(1+i)^n - 1}.
$$

Substitute in the known values:

$$
D = 172000 \cdot \frac{0.00575}{(1.00575)^{132} - 1}.
$$

We first compute $$ (1.00575)^{132} $$. Taking the natural logarithm:

$$
\ln(1.00575) \approx 0.005734,
$$
$$
132 \cdot 0.005734 \approx 0.7563.
$$

Thus,

$$
(1.00575)^{132} \approx e^{0.7563} \approx 2.1307.
$$

Now, subtract 1:

$$
(1.00575)^{132} - 1 \approx 2.1307 - 1 = 1.1307.
$$

Next, compute the numerator:

$$
172000 \times 0.00575 = 989.
$$

Therefore, the deposit is

$$
D \approx \frac{989}{1.1307} \approx 874.33.
$$

Thus, the required equal monthly deposit is approximately

$$
\$874.33.
$$
The required equal monthly deposit is approximately \$874.33.

Acme Annuities recently offered an annuity that pays compounded monthly. What equal monthly deposit should

be made into this annuity in order to have in 11 years?

The amount of each deposit should be

(Round to the nearest cent.)

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Acme Annuities recently offered an annuity that pays compounded monthly. What equal monthly deposit should be made into this annuity in order to have in 11 years? The amount of each deposit should be (Round to the nearest cent.)